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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 16 results · all verified · 14 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Nonlinear parabolic compactness and monotone evolution

1 · Prerequisites

2 · Summary

This page proves continuous and piecewise-constant discrete Aubin–Lions compactness, then uses nonlinear backward Euler Galerkin steps, energy bounds, weak space-time compactness and Minty's argument to identify a subsequential limit for coercive monotone diffusion. The Sobolev state space is $W_{0}^{1, p} (Ω)$ on any bounded open domain, including $d = 1$ .

The concrete $H = L^{2} (Ω)$ branch uses $V = W_{0}^{1, p} (Ω)$ with $p > \max (1, 2 d / (d + 2))$ . In $d = 1$ , compactness follows from zero extension, the one-dimensional fundamental theorem and Arzelà–Ascoli; in $d \geq 2$ , the proof reduces to a subcritical exponent on an arbitrary bounded open set. The page states the Axiom of Choice required by its compact Sobolev embedding and weak compactness interfaces. The energy identity and continuous $H$ trace follow from $u \in L^{p} (0, T; V)$ and $u^{'} \in L^{p^{'}} (0, T; V^{*})$ for every $p > 1$ ; no prior $L^{\infty} (H)$ assumption is needed.

The FEM convergence theorem uses a uniformly $V$ -bounded $H$ -orthogonal projector that converges strongly on $V$ . Local mass-matrix decay and a boundary-preserving quasi-interpolant prove this interface for fixed-degree globally quasi-uniform shape-regular simplex meshes in $d \geq 2$ and directly defined interval partitions in $d = 1$ . Uniform Kuhn/Freudenthal meshes of a unit cube and uniform meshes of a unit interval give explicit nonempty instances. Spatial and time refinement are independent; arbitrary conforming families still meet the abstract projector hypotheses explicitly.

Theorem paths assume the literal hypotheses of parabolic-finite-element-methods for the Hilbert evolution convention and homology-axioms-degree-and-classical-applications only for its published Brouwer fixed-point theorem. The nonlinear flux existence, compactness and identification arguments are proved locally.

The quantitative residual estimate is separate and assumes a strong $p$ -monotonicity inequality. No smooth error rate is asserted for degenerate porous-medium fronts.

Companion examples: nonlinear-parabolic-compactness-and-monotone-evolution-examples.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Banach evolution triples and weak time derivatives

Definition

Let $V$ be a separable reflexive real Banach space, let $H$ be a real Hilbert space, and let $i : V \to H$ be an injective continuous linear map with dense range. Let $V^{*}$ be the continuous real dual of $V$ and write $⟨ \cdot, \cdot ⟩_{V^{*}, V}$ for its pairing with $V$ . The canonical map $J : H \to V^{*}$ is $J h (v) = (h, i v)_{H} (h \in H, v \in V) .$ It is continuous because $∣ J h (v) ∣ \leq ∥ h ∥_{H} ∥ i ∥ ∥ v ∥_{V}$ , and injective because $i (V)$ is dense in $H$ . A triple $(V, H, J)$ with these properties is a Banach evolution triple. If $V$ is Hilbert, this is the real evolution triple of Evolution triples and the parabolic energy space.

For $1 < p < \infty$ , $L^{p} (0, T; V)$ denotes the Bochner space of strongly measurable $V$ -valued functions with finite $p$ th power norm, and $p^{'} = p / (p - 1)$ . For $u \in L^{p} (0, T; V)$ and $g \in L^{p^{'}} (0, T; V^{*})$ , say $g = u^{'}$ in the distributional time sense if $\int_{0}^{T} (i u (t), i v)_{H} ϕ^{'} (t) d t = - \int_{0}^{T} ⟨ g (t), v ⟩_{V^{*}, V} ϕ (t) d t$ for every $v \in V$ and $ϕ \in C_{c}^{\infty} (0, T)$ . Define $W_{p} (0, T; V) = {u \in L^{p} (0, T; V) : u^{'} \in L^{p^{'}} (0, T; V^{*})} .$ This definition assigns no endpoint values; a separate chain-rule theorem is needed before evaluating a function in this space at an endpoint.

Remarks

The published parabolic FEM definition uses $p = 2$ and a Hilbert energy space. This definition allows the reflexive Banach energy space needed for the $p$ -Laplacian branch. The pairing convention is real and bilinear throughout.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The subcritical W0 one p embedding into L2

Statement

Assume the Axiom of Choice. Let $d \geq 1$ , let $Ω \subseteq R^{d}$ be a bounded open set, and let $1 \max (1, 2 d / (d + 2))$ . Then $W_{0}^{1, p} (Ω)$ is continuously and compactly embedded in $L^{2} (Ω)$ . In particular, the inclusion is dense. No boundary regularity is required.

Facts & Assumptions

Given: The Axiom of Choice; a bounded open set $Ω \subseteq R^{d}$ ; $d \geq 1$ ; and $1 \max (1, 2 d / (d + 2))$ .

[F1]

For $d \geq 2$ , if $1 \leq r < d$ and $2 < d r / (d - r)$ , then $W_{0}^{1, r} (Ω)$ is compactly embedded in $L^{2} (Ω)$ for every bounded open $Ω$ , without boundary regularity (Subcritical compactness for $W_{0}^{1, p}$ on arbitrary bounded open sets).

[F2]

For $d \geq 2$ and $1 \leq r < d$ , the zero-boundary Sobolev inequality is $∥ v ∥_{L^{r^{*}}} \leq C ∥ \nabla v ∥_{L^{r}}$ , where $r^{*} = d r / (d - r)$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets).

[F3]

On a finite-measure set, Hölder gives $∥ g ∥_{L^{s}} \leq ∣ Ω ∣^{1 / s - 1 / r} ∥ g ∥_{L^{r}}$ when $1 \leq s < r < \infty$ (Holder's inequality for integrals, including the endpoint cases).

[F4]

For any open $Ω$ , zero extension maps $W_{0}^{1, p} (Ω)$ isometrically into $W^{1, p} (R^{d})$ , with weak gradient the zero extension of $\nabla v$ (Zero extension of W_0^{1,p} has no boundary derivative).

[F5]

For every open $Ω \subseteq R^{d}$ , $C_{c}^{\infty} (Ω)$ is dense in $L^{2} (Ω)$ ; if $Ω \subset R$ is bounded, then $∣ Ω ∣ < \infty$ and uniform convergence on a fixed compact interval implies $L^{2} (Ω)$ convergence (Smooth compactly supported functions of an open set are dense in $L^{2}$ , Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F6]

For a differentiable real function on a compact interval, the fundamental theorem gives the endpoint difference as the integral of the derivative; Hölder bounds that integral by the interval length to the power $1 - 1 / p$ times the derivative's $L^{p}$ norm (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, Holder's inequality for integrals, including the endpoint cases).

[F7]

Under Countable Choice and Dependent Choice, a uniformly bounded equicontinuous family in $C ([a, b]; R)$ has compact closure in the supremum norm (Arzelà--Ascoli for real $C (K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded).

[F8]

$W_{0}^{1, p} (Ω)$ is the norm closure of $C_{c}^{\infty} (Ω)$ in $W^{1, p}$ , and the Sobolev norm includes the function and weak first derivatives (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[A1]

The Axiom of Choice supplies Countable Choice and Dependent Choice; these are used by [F1], [F4], [F5], and [F7] (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F9]

A compact embedding is a continuous inclusion whose image of each bounded set has compact closure; under the stated choice assumptions this is equivalent to the sequential subsequence form used below (Compactly embedded normed spaces).

Proof

1.1A1F4F6F8givenconstruct

If $Ω = \emptyset$ , then $W_{0}^{1, p} (Ω) = L^{2} (Ω) = {0}$ and the conclusion holds. Assume $Ω \neq \emptyset$ . First consider $d = 1$ . Choose a bounded open interval $I = (a, b)$ with $\overline{Ω} \subset I$ and let $L = b - a$ . By [F4] and the closure definition [F8], each $v \in W_{0}^{1, p} (Ω)$ has zero extension $E_{0} v ∣_{I} \in W_{0}^{1, p} (I)$ : choose $φ_{m} \in C_{c}^{\infty} (Ω)$ converging to $v$ in $W^{1, p}$ and extend each by zero into $C_{c}^{\infty} (I)$ . Countable Choice selects this approximating sequence. For every such smooth extension $φ_{m}$ , [F6] gives $∣ φ_{m} (x) - φ_{m} (y) ∣ \leq ∣ x - y ∣^{1 - 1 / p} ∥ φ_{m}^{'} ∥_{L^{p} (I)}$ and $∥ φ_{m} ∥_{L^{\infty} ([a, b])} \leq L^{1 - 1 / p} ∥ φ_{m}^{'} ∥_{L^{p} (I)}$ , because it vanishes near both endpoints. Applying the second inequality to differences shows that $(φ_{m})$ is uniformly Cauchy; its continuous uniform limit $w$ represents $E_{0} v ∣_{I}$ since $φ_{m} \to E_{0} v$ in $L^{p} (I)$ . For every $ζ \in C_{c}^{\infty} (I)$ , uniform convergence and $φ_{m}^{'} \to E_{0} (v^{'})$ in $L^{p} (I)$ give $\int_{I} w ζ^{'} = - \int_{I} E_{0} (v^{'}) ζ$ . Thus $w^{'} = E_{0} (v^{'})$ weakly and [F4] gives $∥ w^{'} ∥_{L^{p} (I)} = ∥ v^{'} ∥_{L^{p} (Ω)}$ . Passing to the limit in the two displayed inequalities gives the same Hölder and supremum bounds for $w$ .

1.2A1F1F3F8givenalgebra

Now let $d \geq 2$ . Put $s_{0} = \max (1, 2 d / (d + 2))$ and $s_{1} = \min (p, d)$ . The strict hypothesis gives $s_{0} < s_{1}$ ; set $r = (s_{0} + s_{1}) / 2$ . Then $1 < r < p$ , $r < d$ , and $2 < r^{*} = d r / (d - r)$ . By finite-measure Hölder [F3], $W_{0}^{1, p} (Ω) ↪ W_{0}^{1, r} (Ω)$ continuously: the function and gradient $L^{r}$ norms are bounded by their $L^{p}$ norms, and a $C_{c}^{\infty}$ approximating sequence in $W^{1, p}$ also converges in $W^{1, r}$ . Therefore [F1] gives compactness of $W_{0}^{1, p} (Ω) \to L^{2} (Ω)$ .

2.1A1F5F7step 1.1givenalgebra

For a bounded sequence $(v_{j})$ in $W_{0}^{1, p} (Ω)$ , the representatives $w_{j}$ from step 1.1 are uniformly bounded by $L^{1 - 1 / p} \sup_{j} ∥ \nabla v_{j} ∥_{p}$ and share the modulus $∣ w_{j} (x) - w_{j} (y) ∣ \leq ∣ x - y ∣^{1 - 1 / p} \sup_{j} ∥ \nabla v_{j} ∥_{p}$ . Arzelà–Ascoli [F7] gives a uniformly convergent subsequence on the fixed compact interval $[a, b]$ . Its restrictions converge in $L^{2} (Ω)$ by [F5], proving compactness. For each $v \in W_{0}^{1, p} (Ω)$ the same supremum estimate gives $∥ v ∥_{2} \leq ∣ Ω ∣^{1 / 2} L^{1 - 1 / p} ∥ \nabla v ∥_{p} \leq L^{3 / 2 - 1 / p} ∥ \nabla v ∥_{p}$ , proving continuity.

3.1F2F3F5F8F9step 1.1step 2.1step 1.2algebra∎

In the $d \geq 2$ branch, for $v \in W_{0}^{1, p} (Ω)$ , [F2] and [F3] yield the continuous estimate $∥ v ∥_{2} \leq ∣ Ω ∣^{1 / 2 - 1 / r^{*}} ∥ v ∥_{r^{*}} \leq C ∣ Ω ∣^{1 / 2 - 1 / r^{*} + 1 / r - 1 / p} ∥ \nabla v ∥_{p}$ , using the Sobolev inequality [F2] with exponent $r$ . Thus the inclusion is continuous on the $W_{0}^{1, p}$ norm. Finally [F5] and the closure definition [F8] give $C_{c}^{\infty} (Ω) \subset W_{0}^{1, p} (Ω)$ dense in $L^{2} (Ω)$ . The preceding arguments prove the complete $d = 1$ and $d \geq 2$ branches; [F9] then gives the claimed compact embedding in its continuous/sequential sense.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Time translations and mollification in Bochner Lp

Statement

Assume Countable Choice.

Let $X$ be a real Banach space, $0 < T < \infty$ , and $1 \leq p < \infty$ . For $f \in L^{p} (0, T; X)$ , translations are continuous in $L^{p}$ on every compact subinterval of $(0, T)$ . Convolution with a scalar smooth approximate identity therefore converges to $f$ in $L^{p}$ on such subintervals. If $P_{k}$ are interval partitions of $[0, T]$ with maximum cell length tending to zero, the piecewise-constant interval averages of $f$ converge to $f$ in $L^{p} (0, T; X)$ .

Facts & Assumptions

Given: Countable Choice, a real Banach space $X$ , $0 < T < \infty$ , $1 \leq p < \infty$ , and $f \in L^{p} (0, T; X)$ .

[F1]

Write $L^{p} (0, T; X)$ for a.e.-classes of strongly measurable functions with $\int_{0}^{T} ∥ f (t) ∥_{X}^{p} d t < \infty$ . Choose the representative to be zero on its exceptional null set; the sets $E_{N}$ below are measurable by [F3]. Strong measurability supplies simple functions $u_{j} \to f$ a.e. For $N \geq 1$ , let $E_{N} = {t : ∥ f (t) ∥_{X} \leq N}$ and $s_{N, j} = u_{j} 1_{E_{N} \cap {∥ u_{j} ∥_{X} \leq N + 1}}$ . These are simple, converge a.e. to $f 1_{E_{N}}$ as $j \to \infty$ , and $∥ s_{N, j} - f 1_{E_{N}} ∥_{X} \leq (2 N + 1) 1_{E_{N}}$ ; dominated convergence on the finite interval gives convergence in $L^{p}$ . Also $f 1_{E_{N}^{c}} \to 0$ in $L^{p}$ by dominated convergence. Hence simple functions are dense in $L^{p} (0, T; X)$ . Since $∥ g ∥_{L^{1}} \leq T^{1 - 1 / p} ∥ g ∥_{L^{p}}$ , this approximation also gives an integrable-simple $L^{1}$ approximation, so $f$ is Bochner integrable by its definition. The norm and integral conventions are those of Strongly measurable Banach-valued function and Bochner-integrable function, with the density argument using Dominated convergence and Holder's inequality for integrals, including the endpoint cases.

[F2]

Scalar translations are continuous in $L^{p} (R)$ for $1 \leq p < \infty$ under Countable Choice ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

Under Countable Choice every Lebesgue measurable set is a Borel set up to a subset of a Borel null set, and a Lebesgue measurable set has an open superset with arbitrarily small excess ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets, For a Lebesgue measurable set and every positive $ε$ there is an open superset whose difference from it has outer measure below $ε$ ). Measures are continuous from below (Continuity from below for measures).

[F4]

For Bochner integrable $v$ , $∥ \int v ∥_{X} \leq \int ∥ v ∥_{X}$ (Bochner integral norm inequality). Scalar Hölder and Minkowski apply to its norm, and scalar integral Minkowski gives $∥ \int F (\cdot, s) d s ∥_{p} \leq \int ∥ F (\cdot, s) ∥_{p} d s$ for nonnegative measurable $F$ with finite right side (Holder's inequality for integrals, including the endpoint cases, Minkowski's inequality for integrals, including $p = \infty$ , Minkowski's integral inequality).

[F5]

By A smooth bump between concentric Euclidean balls, choose a smooth $0 \leq ψ \leq 1$ on $R$ with $ψ = 1$ on $[- 1 / 2, 1 / 2]$ and $supp ψ \subset (- 1, 1)$ ; support is defined in The support of a function on $R^{n}$ and its compactly supported Riemann integral. For $δ > 0$ , $m_{δ} : = \int_{R} ψ (s / δ) d s$ satisfies $δ \leq m_{δ} \leq 2 δ$ , by the interval measure formula and monotonicity of the integral (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Monotonicity and nonnegative homogeneity of the nonnegative integral). Thus $ρ_{δ} (s) : = m_{δ}^{- 1} ψ (s / δ)$ is nonnegative, smooth, supported in $[- δ, δ]$ , and has integral one.

[A1]

Countable Choice is used by the scalar translation supplier [F2] and the completed-measure suppliers [F3]; the simple-function density argument above is derived locally from strong measurability and dominated convergence (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F2givenalgebra

Let $s (t) = \sum_{j = 1}^{m} x_{j} 1_{E_{j}} (t)$ be a simple function with disjoint measurable $E_{j}$ . Extend it by zero to $R$ . By [F2], $∥ 1_{E_{j}} (\cdot - h) - 1_{E_{j}} ∥_{L^{p} (R)} \to 0$ for each $j$ ; the triangle inequality and the finite sum show $∥ s (\cdot - h) - s ∥_{L^{p} (R; X)} \to 0$ .

1.2F1F3F4A1givenconstructalgebra

For an interval partition $P$ , let $P_{P} f$ be the average of $f$ on each cell $I$ . By [F4], $∥ ∣ I ∣^{- 1} \int_{I} f ∥_{X} \leq ∣ I ∣^{- 1} \int_{I} ∥ f ∥_{X}$ . For $p > 1$ , scalar Hölder gives $∣ I ∣ ∥ ∣ I ∣^{- 1} \int_{I} f ∥_{X}^{p} \leq \int_{I} ∥ f ∥_{X}^{p}$ ; for $p = 1$ the first inequality gives the same assertion directly. Summing proves contraction on $L^{p}$ . To justify density for arbitrary completed Lebesgue level sets, write $E = B \cup N$ with $B$ Borel and $N$ contained in a Borel null set $Z$ by [F3], after intersecting all sets with $(0, T)$ . The Borel set $B \cup Z$ differs from $E$ by a null set. Choose an open superset $O$ with excess less than $η$ . The bounded open set $O \cap (0, T)$ is the union of the countably many rational-endpoint open intervals contained in it. Enumerate those intervals, and let $J_{m}$ be their first $m$ unions. By continuity from below [F3], $∣ (O \cap (0, T)) ∖ J_{m} ∣ \to 0$ . Thus a finite union of intervals differs from $E$ in measure by less than $2 η$ for large $m$ . Round the finitely many endpoints by continuous linear ramps, whose error is supported in intervals of arbitrarily small total length. It follows that continuous $X$ -valued functions are dense in $L^{p} (0, T; X)$ by applying this construction to the finitely many level sets of a simple approximation [F1]. For continuous $g$ , uniform continuity gives $∥ P_{P} g - g ∥_{L^{p}} \leq T^{1 / p} \sup_{∣ s - t ∣ \leq \max ∣ I ∣} ∥ g (s) - g (t) ∥_{X} \to 0$ . Contraction now yields $∥ P_{P} f - f ∥_{L^{p}} \leq 2 ∥ f - g ∥_{L^{p}} + ∥ P_{P} g - g ∥_{L^{p}}$ , proving the averages assertion.

2.1A1F1F2step 1.1given

Given $ε > 0$ , choose a simple $s$ with $∥ f - s ∥_{L^{p} (0, T; X)} < ε / 3$ by [F1]. Extend both functions by zero outside $(0, T)$ . Translation is an isometry of the whole-line Bochner norm because it preserves Lebesgue measure; step 1.1 therefore gives, for all sufficiently small $h$ , $∥ f (\cdot - h) - f ∥_{L^{p} (R; X)} \leq 2 ∥ f - s ∥_{L^{p}} + ∥ s (\cdot - h) - s ∥_{L^{p}} < ε$ . Restriction to compact subintervals proves the translation assertion.

3.1F3F4F5step 2.1

Extend $f$ by zero to $R$ and use the kernels from [F5]. Replacing the measurable level sets of simple approximants by Borel representatives modulo null sets gives a Borel version by [F3]; composing the simple approximants with $s \mapsto t - s$ shows the Bochner integrand is strongly measurable in $s$ . Its $L^{p} (R)$ section norm is at most $2 ρ_{δ} (s) ∥ f ∥_{L^{p} (R; X)}$ by translation invariance and Minkowski. Scalar integral Minkowski [F4] makes the $s$ -integral of this norm finite in $L^{p}$ , so the convolution is defined for a.e. $t$ ; by linearity of the defining simple-function limit and $\int ρ_{δ} = 1$ , $ρ_{δ} * f (t) - f (t) = \int_{R} ρ_{δ} (s) (f (t - s) - f (t)) d s$ there. The Bochner norm inequality [F4] then gives $∥ ρ_{δ} * f - f ∥_{L^{p} (R; X)} \leq \int ρ_{δ} (s) ∥ f (\cdot - s) - f ∥_{L^{p} (R; X)} d s$ . Step 2.1 makes the translation modulus uniformly small for $∣ s ∣ \leq δ$ , so this bound tends to zero; restriction to compact subintervals proves the stated mollification convergence.

4.1step 1.2step 2.1step 3.1∎

Steps 1.2, 2.1, and 3.1 prove convergence of interval averages, translation continuity, and convergence of time mollifiers, respectively.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The zero-boundary W1p space is separable and reflexive

Statement

Assume the ultrafilter lemma, Dependent Choice and Hahn–Banach. Let $Ω \subseteq R^{d}$ be bounded and open, $d \geq 1$ , and let $1 < p < \infty$ . Then $W_{0}^{1, p} (Ω; R)$ , with the norm inherited from $W^{1, p} (Ω; R)$ , is a separable reflexive real Banach space.

Facts & Assumptions

Given: The ultrafilter lemma, Dependent Choice and Hahn–Banach; a bounded open $Ω \subseteq R^{d}$ , $d \geq 1$ ; and $1 < p < \infty$ .

[F1]

The displayed $W^{1, p}$ formula is a norm, and weak differentiation is linear; hence $Φ (u) = (u, \partial_{1} u, \dots, \partial_{d} u)$ is a linear embedding of $W^{1, p} (Ω; R)$ into $Y : = L^{p} (Ω; R)^{d + 1}$ with its maximum product norm. Moreover, $∥ Φ u ∥_{\max} \leq ∥ u ∥_{W^{1, p}} \leq (d + 1)^{1 / p} ∥ Φ u ∥_{\max}$ (Integer-order Sobolev spaces and their norms, The Sobolev norm descends to equivalence classes, Linearity, locality, and commutation of weak derivatives).

[F2]

$W_{0}^{1, p} (Ω)$ is the norm closure of $C_{c}^{\infty} (Ω)$ in $W^{1, p} (Ω)$ , so it is a closed linear subspace (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[F3]

Under Hahn–Banach, a closed linear subspace of a reflexive Banach space is reflexive with its inherited norm; reflexive spaces are Banach spaces by definition (Closed subspaces of reflexive spaces are reflexive, Reflexivity is surjectivity of the canonical map).

[F4]

Under Countable Choice, rational box-step functions form a countable dense subset of $L^{p} (R^{d})$ for $1 \leq p < \infty$ (Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ ).

[F5]

A normed space is separable when it has a countable dense subset (Separability: the existence of an at most countable dense subset).

[F6]

Under Countable Choice, $L^{p} (Ω; R)$ is complete for every $1 \leq p \leq \infty$ (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ ).

[F7]

A finite product of Banach spaces is Banach for its maximum product norm (Finite products of Banach spaces are Banach).

[F8]

Under Countable Choice, $L^{p} (Ω; R)$ is reflexive when $1 < p < \infty$ (Reflexivity of Lp for one less p less infinity).

[F9]

For conjugate exponents $p, p^{'}$ , Hölder's inequality bounds each test pairing by the product of the corresponding $L^{p}$ and $L^{p^{'}}$ norms (Holder's inequality for integrals, including the endpoint cases).

[F10]

A compact subset of $R^{d}$ has finite measure, and a continuous function on a compact metric space is bounded (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F11]

Under the ultrafilter lemma, Dependent Choice and Hahn–Banach, a real Banach space is reflexive exactly when every bounded sequence has a weakly convergent subsequence (Reflexivity is equivalent to weak subsequential compactness of bounded sequences).

[F12]

$Q$ is countable and dense in $R$ , subsets and finite products of countable sets are countable, and in a metric space balls centered at a countable dense set with positive rational radii form a countable base. Indeed, for $y \in B (x, r)$ let $δ = r - d (x, y) > 0$ , choose a dense center $z$ with $d (y, z) < δ / 3$ , and choose rational $q$ with $d (y, z) < q < δ - d (y, z)$ ; then $y \in B (z, q) \subseteq B (x, r)$ ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable, Every subset of an at most countable set is at most countable, A product of two at most countable sets is at most countable, Finite, countably infinite, countable, uncountable, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[A1]

The hypotheses include the ultrafilter lemma, Dependent Choice and Hahn–Banach (The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The real dominated-extension principle as an additional hypothesis over ZF). Dependent Choice implies Countable Choice, used by [F4], [F6], [F8] and [F10] (Dependent choice implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F4F12givenconstructchoose

Let $D$ be the countable rational box-step dense family in [F4]. Zero extension embeds $L^{p} (Ω)$ isometrically in $L^{p} (R^{d})$ . Given $g \in L^{p} (Ω)$ , choose $s \in D$ approximating its zero extension $\tilde{g}$ ; then $∥ s ∣_{Ω} - g ∥_{L^{p} (Ω)} \leq ∥ s - \tilde{g} ∥_{L^{p} (R^{d})}$ . Thus the restrictions of $D$ are countable and dense in $L^{p} (Ω)$ . Their finite products give a countable dense subset of $Y$ ; balls centered there with rational radii form a countable base by [F12].

1.2A1F1F6F9F10givenalgebra

Let $(u_{j})$ be Cauchy in $W^{1, p} (Ω)$ . By [F1], the coordinate sequences $(u_{j})$ and $(\partial_{i} u_{j})$ are Cauchy in $L^{p} (Ω)$ for each $i$ . Countable Choice is available by [A1], so [F6] gives limits $u, g_{i} \in L^{p} (Ω)$ . Fix $φ \in C_{c}^{\infty} (Ω)$ with compact support $K$ . By [F10], $K$ has finite measure and $φ, \partial_{i} φ$ are bounded there; Hölder [F9] therefore lets both sides of $\int_{Ω} u_{j} \partial_{i} φ = - \int_{Ω} (\partial_{i} u_{j}) φ$ pass to the limits $u$ and $g_{i}$ . Hence $g_{i} = \partial_{i} u$ weakly for each $i$ , so $u \in W^{1, p} (Ω)$ and $u_{j} \to u$ in its norm by coordinate convergence. Thus $W^{1, p} (Ω)$ is Banach.

2.1A1F1F3F6F7F8F11step 1.2givenconstruct

Each factor of $Y$ is Banach by [F6], so the finite product is Banach for its maximum norm by [F7]. Let $(y_{j})$ be a bounded sequence in $Y$ . Its finitely many coordinate sequences are bounded in $L^{p} (Ω)$ ; by [F8] and the bounded-sequence criterion [F11], successive subsequence extractions give one subsequence weakly convergent in every coordinate. Every bounded linear functional on $Y$ is the sum of its restrictions to the coordinate factors, so this subsequence converges weakly in $Y$ . Thus [F11] makes $Y$ reflexive. By [F1], $Φ$ is a bounded linear isomorphism onto its image $S : = Φ (W^{1, p} (Ω))$ . The image $S$ is closed in $Y$ : a convergent sequence in $S$ has Cauchy preimages by the lower norm bound in [F1], and they converge by step 1.2. Hahn–Banach and [F3] make $S$ reflexive. The inverse isomorphism $Φ^{- 1} : S \to W^{1, p} (Ω)$ preserves weak convergence, so [F11] shows that $W^{1, p} (Ω)$ is reflexive.

3.1A1F2F3step 2.1

By [F2], $W_{0}^{1, p} (Ω)$ is a closed linear subspace of the reflexive Banach space $W^{1, p} (Ω)$ from step 2.1. Hahn–Banach and [F3] therefore make $W_{0}^{1, p} (Ω)$ reflexive and Banach with its inherited norm.

4.1A1F1F5F12step 1.1step 3.1choose

Let $S_{0} = Φ (W_{0}^{1, p} (Ω)) \subseteq Y$ . The countable base from step 1.1 restricts to a countable base of $S_{0}$ . By [A1], Countable Choice selects one point from each nonempty member of this relative base; the resulting countable set meets every nonempty relatively open set and is dense in $S_{0}$ . Since $Φ$ is an isomorphism with equivalent norms by [F1], $W_{0}^{1, p} (Ω)$ is separable by [F5].

5.1step 3.1step 4.1∎

Steps 3.1 and 4.1 establish that $W_{0}^{1, p} (Ω; R)$ is a separable reflexive real Banach space with its inherited norm, as claimed.

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Bochner Lp is complete for a Banach value space

Statement

Let $X$ be a real Banach space, $0 < T < \infty$ , and $1 \leq p < \infty$ . Then $L^{p} (0, T; X)$ , with norm $∥ f ∥_{L^{p} (X)} = (\int_{0}^{T} ∥ f (t) ∥_{X}^{p} d t)^{1 / p}$ , is complete under Countable Choice.

Facts & Assumptions

Given: A real Banach space $X$ , $0 < T < \infty$ , $1 \leq p < \infty$ , and a Cauchy sequence in $L^{p} (0, T; X)$ .

[F1]

Every Cauchy sequence in $X$ has a limit in $X$ (Banach space).

[F2]

Here $L^{p} (0, T; X)$ is the space of a.e.-classes of strongly measurable $X$ -valued functions $f$ with $\int_{0}^{T} ∥ f (t) ∥_{X}^{p} d t < \infty$ , with norm $∥ f ∥_{L^{p} (X)} = (\int_{0}^{T} ∥ f (t) ∥_{X}^{p} d t)^{1 / p}$ . Strong measurability means a.e. pointwise norm convergence of measurable simple functions (Strongly measurable Banach-valued function).

[F3]

After modifying representatives to be zero on exceptional null sets, finite sums and differences of strongly measurable functions are strongly measurable, and their norm functions are measurable: finite sums of simple approximants are simple, while the norm is continuous and pointwise limits of measurable scalar functions are measurable (Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable).

[F4]

The scalar $L^{p}$ norm satisfies Minkowski's inequality. Together with $∥ u (t) + v (t) ∥_{X} \leq ∥ u (t) ∥_{X} + ∥ v (t) ∥_{X}$ , this gives the triangle inequality for the displayed Bochner $L^{p}$ norm (Minkowski's inequality for integrals, including $p = \infty$ ).

[F5]

Increasing nonnegative measurable functions satisfy monotone convergence for their integrals (Monotone convergence for the integral).

[F6]

Dominated convergence holds for nonnegative measurable functions, and the nonnegative integral is monotone (Dominated convergence, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F7]

Measures are monotone and countably subadditive, and a nonnegative extended series is the supremum of its finite partial sums (Measures are monotone, Finite and countable subadditivity of measures, Series in the nonnegative extended real line).

[F8]

The finite interval $(0, T)$ has Lebesgue measure $T$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[A1]

Countable Choice selects representatives and simple-function approximating sequences for the countable subsequence, and indices $m_{k}$ from countably many nonempty approximation sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Subsequence indices can be chosen by least-index recursion.

Proof

1.1A1F3F4F5F6given

The Cauchy sequence has a subsequence $(f_{k})$ with $∥ f_{k + 1} - f_{k} ∥_{L^{p} (X)} \leq 2^{- k}$ for $k \geq 1$ , by recursively taking the least eligible indices. Countable Choice [A1] selects strongly measurable representatives for these classes. Put $g_{N} (t) = \sum_{k = 1}^{N} ∥ f_{k + 1} (t) - f_{k} (t) ∥_{X}$ . By [F3], $g_{N}$ is measurable; Minkowski [F4] gives $∥ g_{N} ∥_{L^{p}} \leq \sum_{k = 1}^{N} 2^{- k} \leq 1$ . Monotone convergence [F5] gives $g (t) = \sum_{k \geq 1} ∥ f_{k + 1} (t) - f_{k} (t) ∥_{X}$ with $\int g^{p} \leq 1$ . Thus $g < \infty$ almost everywhere: for $A = {g = \infty}$ , $M^{p} 1_{A} \leq g^{p}$ for every $M > 0$ , and [F6] gives $M^{p} μ (A) \leq 1$ , forcing $μ (A) = 0$ .

2.1A1F1F2F3F4F6F7F8step 1.1

Outside the null set where $g = \infty$ , the sequence $f_{k} (t)$ is Cauchy in $X$ and converges to $f (t) \in X$ by [F1]; set $f = 0$ on that null set. For each $k$ , choose a simple sequence $s_{k, m} \to f_{k}$ almost everywhere, as supplied by [F2] and [A1]. The sets $E_{k, m} = {t : ∥ s_{k, m} (t) - f_{k} (t) ∥_{X} > 2^{- k}}$ are measurable by [F3], and their indicators tend to zero almost everywhere as $m \to \infty$ . Dominated convergence [F6] on the finite interval [F8] lets Countable Choice [A1] select $m_{k}$ with $μ (E_{k, m_{k}}) \leq 2^{- k}$ . By [F7], the set $N = ⋂_{j \geq 1} ⋃_{k \geq j} E_{k, m_{k}}$ has measure at most $\sum_{k \geq j} 2^{- k} \leq 2^{1 - j}$ for every $j$ , hence is null. Outside $N$ and the previous null set, $s_{k, m_{k}} (t) \to f (t)$ , so $f$ is strongly measurable by [F2]. Also $∥ f - f_{1} ∥_{X} \leq g$ almost everywhere; [F6] makes this difference $L^{p}$ , and Minkowski [F4] gives $f \in L^{p}$ . For each $N$ , $∥ f (t) - f_{N} (t) ∥_{X} \leq \sum_{k \geq N} ∥ f_{k + 1} (t) - f_{k} (t) ∥_{X} \to 0$ and is bounded by $g (t)$ . Dominated convergence [F6] gives $∥ f - f_{N} ∥_{L^{p} (X)} \to 0$ .

3.1F4step 2.1given∎

The original sequence is Cauchy and has the subsequence $f_{N} \to f$ ; given $ε > 0$ , choose $N$ with $∥ f_{N} - f ∥_{L^{p} (X)} < ε / 2$ and then take the original index sufficiently far out that its distance to $f_{N}$ is below $ε / 2$ . The triangle inequality [F4] proves convergence of the entire sequence to $f$ , so $L^{p} (0, T; X)$ is complete.

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An outward-pointing finite-dimensional map has a zero

Statement

Let $N \geq 1$ , $R > 0$ , and let $F : R^{N} \to R^{N}$ be continuous. If $F (x) \cdot x > 0$ for every $∣ x ∣ = R$ , then $F$ has a zero in the closed ball ${\overline{B}}_{R} (0)$ .

Facts & Assumptions

Given: $N \geq 1$ , $R > 0$ , and a continuous $F$ satisfying the strict outward-pointing inequality on the sphere.

[F1]

Every continuous self-map of the closed unit ball in a finite-dimensional real vector space has a fixed point (Brouwer fixed point theorem).

Proof

1.1F1givendischarge-contradictionconstruct

Suppose, for contradiction, that $F$ has no zero in ${\overline{B}}_{R} (0)$ . Then $F (x) \neq 0$ there, and $G (x) : = - R F (x) / ∣ F (x) ∣$ is a continuous map ${\overline{B}}_{R} (0) \to {\overline{B}}_{R} (0)$ whose values all lie on the sphere. After rescaling the ball to the unit ball, [F1] gives a fixed point $x = G (x)$ .

2.1F1step 1.1givendischarge-contradictionalgebra∎

Since $G (x)$ lies on the sphere, $∣ x ∣ = R$ . The fixed-point equation gives $F (x) \cdot x = F (x) \cdot (- R F (x) / ∣ F (x) ∣) = - R ∣ F (x) ∣ < 0$ , contradicting the assumed positive boundary pairing. Thus $F$ has a zero in the closed ball.

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Quasi-uniform finite-element mass inverses have summable graph decay

Statement

Assume Countable Choice. Fix a degree $k \geq 1$ . The mesh contract has two branches. In dimensions $d \geq 2$ , let $Ω \subset R^{d}$ be a bounded polyhedral open set and let $T_{h}$ be finite matching affine simplex meshes as in Matching affine simplicial meshes and shape regularity, with $h = \max_{K \in T_{h}} diam K$ , $h \to 0$ , $diam K \leq h$ , $diam K \geq θ h$ , and $diam K / ρ_{K} \leq σ$ for constants $θ > 0$ and $σ < \infty$ independent of $h$ . In dimension $d = 1$ , let $Ω$ be a bounded interval or a finite union of bounded intervals with pairwise disjoint closures, and let $T_{h}$ be finite interval partitions of the component closures meeting only at whole endpoints, with interval lengths between $θ h$ and $h$ and $h \to 0$ .

For each mesh, place the degree- $k$ barycentric Lagrange nodes on every $d \geq 2$ simplex, or the $k + 1$ equally spaced Lagrange nodes on every $d = 1$ interval, identify coincident nodes of matching cells, and impose zero coefficients at every node on $\partial Ω$ . Let $V_{h}$ be the resulting scalar continuous piecewise- $P_{k}$ space. If $z_{i}$ are its interior nodes, $ϕ_{i}$ the global nodal basis, and

M_{i j} = \int_{Ω} ϕ_{i} ϕ_{j}, A_{h} = h^{- d} M,

join distinct nodes when they occur in one cell and write $δ_{h} (i, j)$ for the shortest path length in this graph (with $δ_{h} (i, j) = \infty$ when there is no path, and use $q^{\infty} = 0$ ). Whenever $V_{h} \neq {0}$ , there are $a, b, C > 0$ and $q \in (0, 1)$ , independent of the mesh, such that $spec (A_{h}) \subset [a, b], ∣ (A_{h}^{- 1})_{i j} ∣ \leq C q^{δ_{h} (i, j)}, \sup_{i} \sum_{j} ∣ (A_{h}^{- 1})_{i j} ∣ + \sup_{j} \sum_{i} ∣ (A_{h}^{- 1})_{i j} ∣ \leq C .$ For every cell $K$ and $1 \leq r < \infty$ , if $c_{K}$ are its local nodal coefficients and $w_{h} = \sum_{i \in I (K)} c_{i} ϕ_{i}$ on $K$ , then $c_{r} h_{K}^{d / r} ∥ c_{K} ∥_{ℓ^{r}} \leq ∥ w_{h} ∥_{L^{r} (K)} \leq C_{r} h_{K}^{d / r} ∥ c_{K} ∥_{ℓ^{r}}, ∥ \nabla w_{h} ∥_{L^{r} (K)} \leq C_{r} h_{K}^{d / r - 1} ∥ c_{K} ∥_{ℓ^{r}} .$ At $r = \infty$ , omit $h_{K}^{d / r}$ in the function estimate and replace the gradient bound by $∥ \nabla w_{h} ∥_{L^{\infty} (K)} \leq C h_{K}^{- 1} ∥ c_{K} ∥_{ℓ^{\infty}}$ . Summing gives, for all global nodal coefficients including zero boundary coefficients, $c_{r} h^{d / r} ∥ c ∥_{ℓ^{r}} \leq ∥ \sum_{i} c_{i} ϕ_{i} ∥_{L^{r} (Ω)} \leq C_{r} h^{d / r} ∥ c ∥_{ℓ^{r}} .$ with the factor $h^{d / r}$ omitted at $r = \infty$ . The empty interior-node case is interpreted as $V_{h} = {0}$ and has no mass matrix to invert.

Facts & Assumptions

Given: Countable Choice; one of the two stated mesh branches; a fixed $k \geq 1$ ; and the scalar continuous degree- $k$ nodal space with zero boundary coefficients.

[F1]

For $d \geq 2$ , a matching affine simplicial mesh consists of finitely many closed nondegenerate simplices with disjoint interiors whose union is $\overline{Ω}$ and whose pairwise intersections are whole faces (Matching affine simplicial meshes and shape regularity). The $d = 1$ interval-partition branch is defined explicitly in this Statement and does not use that $d \geq 2$ definition.

[F2]

On a fixed reference simplex, polynomial $L^{r}$ norms, derivative seminorms, facet norms, and every linear nodal functional are controlled by finite-dimensional norm equivalence (Polynomial derivative and face norms are controlled on a reference simplex). The affine maps of a shape-regular simplex scale cell volume like $h_{K}^{d}$ and derivatives like $h_{K}^{- 1}$ (Shape regularity controls affine simplex geometry).

[F3]

A real symmetric endomorphism on a finite-dimensional real inner-product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F4]

Axis-parallel boxes with real endpoints have Lebesgue measure equal to the product of their side lengths; measures are additive on finite disjoint families and monotone (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Measures on sigma-algebras, Measures are monotone).

[A1]

Countable Choice is used through the Sobolev and finite-dimensional norm interfaces in [F2], and the box-measure result in [F4] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1givenalgebra

On a $d \geq 2$ reference simplex with barycentric coordinates $λ_{0}, \dots, λ_{d}$ , take the nodes $(α_{0} / k, \dots, α_{d} / k)$ for nonnegative integers $α_{j}$ with $\sum_{j} α_{j} = k$ . They are unisolvent for $P_{k}$ . Indeed, use induction on $d + k$ , with the zero-dimensional constant case and the degree-one vertex case as bases. For $k \geq 2$ , restriction to the face $λ_{0} = 0$ vanishes identically by the dimension induction, so the polynomial is $λ_{0} R$ with $\deg R \leq k - 1$ . At nodes with $α_{0} \geq 1$ , the affine change $λ_{0} = (1 + (k - 1) μ_{0}) / k$ and $λ_{j} = (k - 1) μ_{j} / k$ for $j \geq 1$ identifies the remaining nodes with the degree- $(k - 1)$ barycentric lattice. The induction makes $R$ vanish at every node of that lattice, hence $R = 0$ . The number of barycentric nodes is $(\binom{k + d}{d})$ by placing $d$ separators among $k$ stars; this equals the number of degree-at-most- $k$ monomials in $d$ variables, so it is $\dim P_{k}$ and the unisolvent values define a basis. In $d = 1$ , the equally spaced $k + 1$ nodes are unisolvent because a nonzero degree- $k$ polynomial has at most $k$ roots. On a shared face the two cells have the same face nodes; unisolvence of the face restriction makes their polynomial traces agree when the shared nodal values agree. The same face-node unisolvence shows that zero nodal coefficients on a boundary facet give its zero polynomial trace; in $d = 1$ the boundary facets are the zero-valued endpoints. Unisolvence also means that local nodal interpolation reproduces every polynomial in $P_{k}$ , so the local basis sums to one and reproduces affine polynomials. Thus the constructed nodal functions are conforming and have the declared zero boundary values.

1.2F1F2F4givenalgebra

If $δ_{h} (i, j) \leq m$ , a path of $m$ common-cell links gives $∣ z_{i} - z_{j} ∣ \leq m h$ . Any cell containing such a node lies in $B (z_{i}, (m + 1) h)$ . In $d \geq 2$ , quasi-uniformity and shape regularity give $ρ_{K} \geq θ h / σ$ ; each cell therefore contains an axis-parallel open cube of side $2 θ h / (σ \sqrt{d})$ inside an inscribed ball. These cubes are disjoint and lie in the box $(z_{i} - (m + 2) h, z_{i} + (m + 2) h)^{d}$ . By [F4], each has measure at least $(2 θ h / (σ \sqrt{d}))^{d}$ and the outer box has measure $(2 (m + 2) h)^{d}$ , so additivity and monotonicity bound the number of such cells by $C (m + 1)^{d}$ . In $d = 1$ , the intervals lie in an interval of length $2 (m + 1) h$ and have lengths at least $θ h$ , giving the same cell bound. Each cell contains a fixed number $N (d, k)$ of nodes, hence $# {j : δ_{h} (i, j) \leq m} \leq C (d, k) (m + 1)^{d} .$ The same packing bound shows that the number of cells incident to any fixed node is uniformly bounded, since all such cells lie in $B (z_{i}, h)$ .

2.1A1F2step 1.1givenalgebra

On each reference cell, [F2] bounds the $L^{r}$ norm above and below by the $ℓ^{r}$ norm of the finitely many nodal coefficients; it also bounds the reference gradient norm above by that coefficient norm. Affine scaling then gives, for $1 \leq r < \infty$ , $∥ \sum_{i \in I (K)} c_{i} ϕ_{i} ∥_{L^{r} (K)}^{r} ≍ h_{K}^{d} \sum_{i \in I (K)} ∣ c_{i} ∣^{r}, ∥ \nabla \sum_{i \in I (K)} c_{i} ϕ_{i} ∥_{L^{r} (K)} \leq C h_{K}^{d / r - 1} ∥ c ∥_{ℓ^{r} (I (K))} .$ For $r = \infty$ replace the first factor $h_{K}^{d / r}$ by $1$ . The same statements follow on an interval from $x = a + h_{K} t$ and the fixed one-dimensional nodal basis. Since $θ h \leq h_{K} \leq h$ , all cell scales are uniform in $h$ .

3.1step 1.2step 2.1givenalgebra

For finite $r$ , sum the local $L^{r}$ equivalence in step 2.1 over all cells. Each cell contains a fixed number $N (d, k)$ of nodes, and each global node is incident to uniformly many cells by the packing bound in step 1.2 (in $d = 1$ at most two intervals meet at a node). Thus each global coefficient occurs between one and a uniformly bounded number of times, giving the asserted global lower and upper $ℓ^{r}$ scaling. Taking maxima of local polynomial norm equivalences gives the $r = \infty$ scaling.

4.1step 3.1givenalgebra

The interior-node mass matrix is symmetric positive definite because $c^{T} M c = ∥ \sum_{i} c_{i} ϕ_{i} ∥_{L^{2} (Ω)}^{2}$ , which is positive for every nonzero coefficient vector by the lower scaling in step 3.1. Taking $r = 2$ there shows $a ∥ c ∥_{2}^{2} \leq c^{T} A_{h} c \leq b ∥ c ∥_{2}^{2}$ for positive constants independent of the mesh.

5.1F3step 4.1givenalgebra

Put $β = 2 b$ and $B = I - A_{h} / β$ . By step 4.1 and [F3], $∥ B ∥_{2 \to 2} \leq q : = 1 - a / (2 b) < 1$ . The matrix $A_{h}$ has zero $i j$ entry unless $i = j$ or $i, j$ occur in a common cell. Consequently $(B^{n})_{i j} = 0$ for $n < δ_{h} (i, j)$ , since a nonzero product of $n$ entries would give a graph path of length at most $n$ . The norm-convergent Neumann series gives $A_{h}^{- 1} = β^{- 1} \sum_{n = 0}^{\infty} B^{n}, ∣ (A_{h}^{- 1})_{i j} ∣ \leq β^{- 1} \sum_{n \geq δ_{h} (i, j)} q^{n} = \frac{q^{δ_{h} (i, j)}}{β (1 - q)} .$ If $δ_{h} (i, j) = \infty$ , every summand is zero and the inverse entry is zero.

6.1step 1.2step 5.1algebra∎

Grouping the entry bound of step 5.1 by graph distance and applying the ball count of step 1.2 gives each absolute row sum at most $C \sum_{m \geq 0} (m + 1)^{d} q^{m} < \infty$ ; the series converges by the ratio test because its term ratio tends to $q < 1$ . Symmetry of $A_{h}^{- 1}$ gives the same column bound. These constants do not depend on the number of cells.

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Coercive monotone p-growth diffusion fluxes

Definition

Assume the Axiom of Choice (The Axiom of Choice).

Let $d \geq 1$ , let $Ω \subset R^{d}$ be a bounded open set, and let $1 \max (1, 2 d / (d + 2))$ . Put $V = W_{0}^{1, p} (Ω; R)$ with norm $∥ v ∥_{V} = ∥ \nabla v ∥_{L^{p} (Ω)}$ , $H = L^{2} (Ω; R)$ , and $p^{'} = p / (p - 1)$ . The Poincaré inequality makes this norm equivalent to the full $W^{1, p}$ norm. The zero-boundary W1p space is separable and reflexive gives separability and reflexivity, while The subcritical W0 one p embedding into L2 gives the dense continuous compact inclusion $V ⋐ H$ ; these are the hypotheses of the evolution triple in Banach evolution triples and weak time derivatives. The compact-inclusion result uses the stated threshold also when $d = 1$ , where it reduces to $p > 1$ .

A measurable flux $a : Ω \times R^{d} \to R^{d}$ is an admissible coercive monotone diffusion flux of p-growth if it is Carathéodory (for each $ξ$ , $x \mapsto a (x, ξ)$ is measurable, and for almost every $x$ , $ξ \mapsto a (x, ξ)$ is continuous), and there are constants $C < \infty$ , $α > 0$ , nonnegative functions $g \in L^{p^{'}} (Ω)$ and $h \in L^{1} (Ω)$ , and one null set outside which, for all $ξ, η \in R^{d}$ , $∣ a (x, ξ) ∣ \leq C (g (x) + ∣ ξ ∣^{p - 1}), a (x, ξ) \cdot ξ \geq α ∣ ξ ∣^{p} - h (x),$ $(a (x, ξ) - a (x, η)) \cdot (ξ - η) \geq 0.$ The induced operator $A : V \to V^{*}$ is $⟨ A (v), w ⟩_{V^{*}, V} = \int_{Ω} a (x, \nabla v (x)) \cdot \nabla w (x) d x .$ The growth bound and Hölder's inequality make this pairing finite and give $∥ A (v) ∥_{V^{*}} \leq C_{1} (∥ g ∥_{p^{'}} + ∥ v ∥_{V}^{p - 1})$ for a constant $C_{1}$ . The finite-dimensional test-coordinate continuity used in the backward Euler existence argument follows from the stated Carathéodory and growth conditions by dominated convergence, as proved in Nonlinear backward Euler Galerkin existence and energy bounds ↗. No additional global operator-continuity hypothesis is imposed.

For $f \in L^{p^{'}} (0, T; V^{*})$ , a weak evolution solution is a function $u \in W_{p} (0, T; V) \cap L^{\infty} (0, T; H)$ with a continuous $H$ representative such that $u^{'} + A (u) = f$ in $L^{p^{'}} (0, T; V^{*})$ and $u (0) = u_{0} \in H$ .

Remarks

The $L^{2}$ state space is available exactly on the stated strict subcritical branch. The growth, coercivity, and monotonicity conditions are explicit; no smoothness or error rate for degenerate fronts is implicit. No boundary regularity is assumed: the Poincaré and compact embedding suppliers used above hold on bounded open sets.

For $v \in V$ , apply A Caratheodory integrand composed with measurable functions is measurable to each component of $a$ (independent of its scalar argument) and the measurable map $\nabla v$ ; this makes $x \mapsto a (x, \nabla v (x))$ measurable. The growth condition gives $∥ a (\cdot, \nabla v) ∥_{L^{p^{'}} (Ω)} \leq C (∥ g ∥_{L^{p^{'}} (Ω)} + ∥ \nabla v ∥_{L^{p} (Ω)}^{p - 1}),$ because $(p - 1) p^{'} = p$ . Thus Hölder's inequality makes the displayed pairing a bounded linear functional of $w$ , so $A (v) \in V^{*}$ with the stated norm bound.

The choice assumptions of the Sobolev suppliers are met under the stated Axiom of Choice: it implies Countable and Dependent Choice by AC supplies the countable and dependent choices used in Banach integration, the ultrafilter lemma by The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, and Hahn–Banach by Hahn-Banach dominated extension theorem for real vector spaces.

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Conforming FEM families with stable L2 projectors

Definition

Let $(V, H, J)$ be the Banach evolution triple of Banach evolution triples and weak time derivatives. An admissible stable conforming FEM sequence consists of finite-dimensional subspaces $V_{k} \subset V$ and their $H$ -orthogonal projections $Π_{k} : H \to V_{k}$ such that:

$Π_{k} (V) \subset V$ , $Π_{k} v = v$ for $v \in V_{k}$ , and $(w_{k}, z - Π_{k} z)_{H} = 0$ for every $w_{k} \in V_{k}$ , $z \in H$ ;
$\sup_{k} ∥ Π_{k} ∥_{V \to V} \leq C_{Π} < \infty$ and $Π_{k} v \to v$ in $V$ for each $v \in V$ ;
the associated time partitions of $[0, T]$ have maximum step $δ_{k} \to 0$ .

The convergence in clause 2 and density of $V$ in $H$ imply $Π_{k} z \to z$ in $H$ for every $z \in H$ : use $∥ Π_{k} (z - v) ∥_{H} \leq ∥ z - v ∥_{H}$ , $Π_{k} v \to v$ in $H$ , and approximate $z$ by $v \in V$ . Thus the range spaces approximate every $H$ vector.

For a fixed $k$ , backward Euler with averages $f_{k}^{n}$ uses unknowns $U_{k}^{n} \in V_{k}$ and equations ${(\frac{U_{k}^{n} - U_{k}^{n - 1}}{τ_{n}}, v_{k})}_{H} + ⟨ A (U_{k}^{n}), v_{k} ⟩ = ⟨ f_{k}^{n}, v_{k} ⟩ (v_{k} \in V_{k}) .$ Its piecewise-constant reconstruction is ${\overline{U}}_{k} (t) = U_{k}^{n}$ on $(t_{n - 1}, t_{n}]$ ; its affine reconstruction joins $U_{k}^{n - 1}$ to $U_{k}^{n}$ .

Remarks

The uniform $V$ -stability of the $H$ -orthogonal projections is an explicit method hypothesis; conformity alone does not imply it. The same-page L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes proves this interface for fixed-degree globally quasi-uniform shape-regular simplex meshes on bounded Lipschitz polyhedral domains in $d \geq 2$ and for the directly defined interval meshes in $d = 1$ . The abstract definition remains applicable to other conforming families when its projector clauses are independently verified.

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The Banach evolution-space energy identity and H trace

Statement

Assume the Axiom of Choice. Let $(V, H, J)$ be the Banach evolution triple of Banach evolution triples and weak time derivatives, let $1 < p < \infty$ , and suppose $u \in L^{p} (0, T; V)$ with distributional derivative $u^{'} \in L^{p^{'}} (0, T; V^{*})$ . Then $u$ has a unique representative in $C ([0, T]; H)$ and, for every $0 \leq s \leq t \leq T$ , $∥ u (t) ∥_{H}^{2} - ∥ u (s) ∥_{H}^{2} = 2 \int_{s}^{t} ⟨ u^{'} (r), u (r) ⟩_{V^{*}, V} d r .$

Facts & Assumptions

Given: The Axiom of Choice, the Banach evolution triple, $1 < p < \infty$ , $u \in L^{p} (0, T; V)$ , and $u^{'} \in L^{p^{'}} (0, T; V^{*})$ .

[F1]

The distributional derivative is defined by scalar tests against each $v \in V$ and $ϕ \in C_{c}^{\infty} (0, T)$ , with the embedding $V \to H$ and canonical map $H \to V^{*}$ (Banach evolution triples and weak time derivatives).

[F2]

Bochner time mollification converges in $L^{r} (X)$ on compact interior subintervals for every real Banach space $X$ and $1 \leq r < \infty$ (Time translations and mollification in Bochner Lp). Derivative commutation is proved in step 1.1 below.

[F3]

The inclusion $V \to H$ is continuous, $H$ is complete, and its inner product satisfies the norm identity used by the ordinary product rule (Banach evolution triples and weak time derivatives, Hilbert space).

[F4]

Hölder's inequality makes every $L^{p^{'}} (V^{*})$ -- $L^{p} (V)$ pairing integrable (Holder's inequality for integrals, including the endpoint cases).

[F5]

A scalar one-dimensional $W^{1, 1} (0, T)$ class has an absolutely continuous representative on $[0, T]$ (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives).

[A1]

The Axiom of Choice supplies Countable and Dependent Choice for the Bochner approximation and endpoint selections (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

1.1A1F1F2F3F4givenconstruct

Fix a compact interval $K \subset (0, T)$ and a smooth unit-mass kernel $ρ_{ϵ}$ supported in $(- ϵ, ϵ)$ , with $ϵ < dist (K, {0, T})$ . Set $g = u^{'}$ and $u_{ϵ} (t) = \int_{0}^{T} ρ_{ϵ} (t - r) u (r) d r$ . This is smooth in $V$ : difference quotients of the kernel and each of its derivatives converge uniformly on the compact integration region, and multiplying their uniform error by $∥ u ∥_{L^{1} (V)} < \infty$ bounds the error in the differentiated Bochner integral. For $v \in V$ , use the scalar test $r \mapsto ρ_{ϵ} (t - r)$ in [F1]; since its derivative is $- ρ_{ϵ}^{'} (t - r)$ , this gives $⟨ J i {\dot{u}}_{ϵ} (t), v ⟩ = \int_{0}^{T} ρ_{ϵ}^{'} (t - r) (i u (r), i v)_{H} d r = \int_{0}^{T} ρ_{ϵ} (t - r) ⟨ g (r), v ⟩ d r .$ Thus the pivot derivative $u_{ϵ}^{'} : = J i {\dot{u}}_{ϵ}$ equals $ρ_{ϵ} * g$ in $V^{*}$ . The ordinary $H$ product rule now yields $\frac{d}{d t} \frac{1}{2} ∥ i u_{ϵ} (t) ∥_{H}^{2} = ⟨ u_{ϵ}^{'} (t), u_{ϵ} (t) ⟩ .$ Applying [F2] separately with $(X, r) = (V, p)$ and $(V^{*}, p^{'})$ proves $u_{ϵ} \to u$ in $L^{p} (K; V)$ and $u_{ϵ}^{'} \to u^{'}$ in $L^{p^{'}} (K; V^{*})$ . Below, $V$ -valued functions in $H$ expressions are identified with their images under $i$ .

2.1A1F2F3F4step 1.1choosealgebra

For two small radii $ϵ, δ$ , put $w = u_{ϵ} - u_{δ}$ on a fixed nondegenerate compact interval $K \subset (0, T)$ . Continuity gives a time $s \in K$ with $∥ w (s) ∥_{H}^{p} \leq ∣ K ∣^{- 1} ∥ w ∥_{L^{p} (K; H)}^{p}$ : otherwise integration contradicts the definition of that norm. Apply the smooth energy identity of step 1.1 to the difference $w$ and integrate between $s$ and any $t \in K$ . Hölder [F4] yields $\sup_{t \in K} ∥ w (t) ∥_{H}^{2} \leq ∣ K ∣^{- 2 / p} ∥ w ∥_{L^{p} (K; H)}^{2} + 2 ∥ w^{'} ∥_{L^{p^{'}} (K; V^{*})} ∥ w ∥_{L^{p} (K; V)} ⟶ 0.$ Indeed [F2] makes both $w$ and $w^{'}$ tend to zero in their indicated norms, and the continuous inclusion [F3] controls the $H$ -valued $L^{p}$ norm. This holds for every $p > 1$ without a prior $L^{\infty} (H)$ bound. Thus the mollifications are uniformly Cauchy in $H$ on $K$ .

3.1F2F4step 1.1step 2.1algebra

Completeness of $H$ gives a uniform limit in $C (K; H)$ , which agrees almost everywhere with $u$ because $u_{ϵ} \to u$ in $L^{p} (K; H)$ . The limits agree on overlapping compact intervals, so they define $u \in C ((0, T); H)$ . Integrate step 1.1 between arbitrary $s, t \in K$ . Uniform convergence passes the endpoint norms to those of $u$ , while $∥ ⟨ u_{ϵ}^{'}, u_{ϵ} ⟩ - ⟨ u^{'}, u ⟩ ∥_{L^{1} (K)} \leq ∥ u_{ϵ}^{'} - u^{'} ∥_{L^{p^{'}} (V^{*})} ∥ u_{ϵ} ∥_{L^{p} (V)} + ∥ u^{'} ∥_{L^{p^{'}} (V^{*})} ∥ u_{ϵ} - u ∥_{L^{p} (V)} \to 0.$ Hence for all $0 < s \leq t < T$ , $∥ u (t) ∥_{H}^{2} - ∥ u (s) ∥_{H}^{2} = 2 \int_{s}^{t} ⟨ u^{'} (r), u (r) ⟩ d r .$

4.1F1F4F5step 3.1

By [F4], the energy pairing is in $L^{1} (0, T)$ . The identity in step 3.1 therefore shows that $∥ u (t) ∥_{H}^{2}$ has finite limits as $t ↓ 0$ and $t ↑ T$ . For each fixed $v \in V$ , the distributional derivative of $(u (t), i v)_{H}$ is $⟨ u^{'} (t), v ⟩ \in L^{1} (0, T)$ , so [F5] gives an absolutely continuous representative on $[0, T]$ agreeing with the interior pairing.

5.1A1F4F5step 3.1step 4.1givenchoose

For $0 < δ < T$ , choose a good time $s_{δ} \in (0, δ)$ with $u (s_{δ}) \in V$ and $∥ u (s_{δ}) ∥_{V}^{p} \leq \frac{2}{δ} \int_{0}^{δ} ∥ u (r) ∥_{V}^{p} d r .$ Such a time exists by averaging over $(0, δ)$ . For every interior $t \in (0, δ)$ , use step 3.1 for the two norm squares and [F5] for the scalar pairing against the fixed vector $u (s_{δ}) \in V$ . Subtracting twice that pairing identity gives $∥ u (t) - u (s_{δ}) ∥_{H}^{2} = 2 \int_{s_{δ}}^{t} ⟨ u^{'} (r), u (r) - u (s_{δ}) ⟩ d r .$ Hölder and the triangle inequality bound its absolute value by $2 ∥ u^{'} ∥_{L^{p^{'}} (0, δ; V^{*})} (∥ u ∥_{L^{p} (0, δ; V)} + δ^{1 / p} ∥ u (s_{δ}) ∥_{V}),$ which tends to zero as $δ ↓ 0$ . The bound is uniform in $t \in (0, δ)$ , so $u (t)$ is Cauchy in $H$ as $t ↓ 0$ and has a strong limit there. The same argument on $(T - δ, T)$ gives a strong limit at $T$ . Thus $u$ extends to $C ([0, T]; H)$ .

6.1F4step 3.1step 5.1given∎

Let $s ↓ 0$ and $t ↑ T$ in step 3.1. Strong endpoint continuity and integrability of $⟨ u^{'}, u ⟩$ pass the identity to all $0 \leq s \leq t \leq T$ . If two continuous representatives agree almost everywhere, their difference is continuous and vanishes on a dense subset, so it vanishes everywhere. This proves uniqueness and the claimed chain rule.

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Aubin–Lions compactness with exact time exponent

Statement

Assume the Axiom of Choice. Let $(V, H, J)$ be a Banach evolution triple with compact inclusion $i : V ⋐ H$ and continuous canonical inclusion $J : H ↪ V^{*}$ . Let $T < \infty$ , $1 < p < \infty$ , and $1 < q \leq \infty$ . Suppose $u_{j} \in L^{p} (0, T; V)$ have distributional derivatives $u_{j}^{'} \in L^{q} (0, T; V^{*})$ and $\sup_{j} (∥ u_{j} ∥_{L^{p} (0, T; V)} + ∥ u_{j}^{'} ∥_{L^{q} (0, T; V^{*})}) < \infty .$ Then $(u_{j})$ is relatively compact in $L^{p} (0, T; H)$ . If also $\sup_{j} ∥ u_{j} ∥_{L^{\infty} (0, T; H)} < \infty$ , each subsequence converging in $L^{p} (0, T; H)$ converges, after the same subsequence, in every $L^{r} (0, T; H)$ with $1 \leq r < \infty$ .

Facts & Assumptions

Given: The Axiom of Choice, the Banach evolution triple, compact and continuous embeddings, $T < \infty$ , $1 < p < \infty$ , $1 < q \leq \infty$ , and the stated uniform bounds.

[F1]

Compact embedding means every bounded sequence in $V$ has an $H$ -convergent subsequence; the canonical map $J$ is continuous and $⟨ J h, v ⟩ = (h, i v)_{H}$ (Compactly embedded normed spaces, Banach evolution triples and weak time derivatives).

[F2]

In this theorem, $g = u_{j}^{'} \in L^{q} (0, T; V^{*})$ means $\int_{0}^{T} (i u_{j} (t), i v)_{H} ϕ^{'} (t) d t = - \int_{0}^{T} ⟨ g (t), v ⟩ ϕ (t) d t$ for every $v \in V$ and $ϕ \in C_{c}^{\infty} (0, T)$ . This is the scalar-test convention of Banach evolution triples and weak time derivatives, here extended to the stated independent exponent $q$ ; membership in $L^{p^{'}} (V^{*})$ is not required. Both integrals exist by the finite-interval $L^{1}$ bounds from [F3]. In particular, $t \mapsto (i u_{j} (t), i v)_{H}$ has scalar weak derivative $t \mapsto ⟨ g (t), v ⟩$ .

[F3]

Hölder and Minkowski control scalar pairings, integrals and finite Bochner sums (Holder's inequality for integrals, including the endpoint cases, Minkowski's inequality for integrals, including $p = \infty$ ).

[F4]

A scalar $W^{1, q} (0, T)$ class, $1 \leq q \leq \infty$ , has an absolutely continuous representative on $[0, T]$ (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives).

[F5]

The Bochner space $L^{p} (0, T; H)$ is complete (Bochner Lp is complete for a Banach value space).

[A1]

The Axiom of Choice supplies Countable and Dependent Choice for sequential extraction and countable dense-set selections (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

1.1A1F1givenchoose

Fix $ε > 0$ . Compactness in [F1] makes the image in $H$ of the $V$ unit ball totally bounded. Take a finite $ε / 2$ -net in $H$ for this image; since the range $i (V)$ is dense in $H$ by [F1], approximate each net point within $ε / 2$ by a point of $i (V)$ . Let $E \subset i (V)$ be the span of these finitely many points. The $H$ -orthogonal projection $P_{E}$ is the best approximation from $E$ , so $∥ (I - P_{E}) i v ∥_{H} \leq ε ∥ v ∥_{V} (v \in V) .$ After orthonormalizing a basis, write $e_{ℓ} = i v_{ℓ}$ with $v_{ℓ} \in V$ .

1.2A1F2F3F4given

For each $j, ℓ$ , put $a_{j ℓ} (t) = (u_{j} (t), e_{ℓ})_{H}$ . Its weak derivative is $⟨ u_{j}^{'} (t), v_{ℓ} ⟩$ by [F2]. The uniform bounds and the continuous $V \to H$ inclusion give uniform $L^{p}$ bounds on $a_{j ℓ}$ and uniform $L^{q}$ bounds on their derivatives. Since $T < \infty$ , each coefficient is in $L^{1} (0, T)$ and its derivative is in $L^{1} (0, T)$ ; hence it is in scalar $W^{1, 1} (0, T)$ and [F4] gives an absolutely continuous representative. For any such representative, some $s \in [0, T]$ has $∣ a_{j ℓ} (s) ∣ \leq T^{- 1 / p} ∥ a_{j ℓ} ∥_{L^{p}}$ ; otherwise its $L^{p}$ norm would exceed that bound. Its total variation is at most $T^{1 - 1 / q} ∥ a_{j ℓ}^{'} ∥_{L^{q}}$ (with $T^{1 - 1 / q} = T$ for $q = \infty$ ). Thus all coefficients are uniformly bounded, and Hölder gives the common modulus $∣ a_{j ℓ} (t) - a_{j ℓ} (s) ∣ \leq C ∣ t - s ∣^{1 - 1 / q}$ for $q < \infty$ , with $C ∣ t - s ∣$ for $q = \infty$ .

2.1A1F3step 1.2choose

The finitely many coefficient functions of $P_{E} u_{j}$ are uniformly bounded and share the modulus from step 1.2. Choose a finite time grid so the modulus between adjacent grid points is below $ε$ , and finite $ε$ -nets for their bounded coefficient vectors at each grid point. The piecewise-linear interpolants of the finitely many rounded grid values form a finite family; each coefficient vector is within $2 ε$ of its corresponding interpolant in the uniform norm, by the common modulus and the grid-point rounding error. Hence ${P_{E} u_{j}}$ is totally bounded in $L^{p} (0, T; H)$ .

3.1F3step 1.1step 2.1algebra

By step 1.1, $∥ u_{j} - P_{E} u_{j} ∥_{L^{p} (0, T; H)} \leq ε ∥ u_{j} ∥_{L^{p} (0, T; V)} .$ For each desired accuracy, choose $ε$ small and use step 2.1 to approximate the projected family by a finite net. The displayed remainder then shows that the full family is totally bounded in $L^{p} (0, T; H)$ .

4.1A1F3F5step 3.1algebra∎

From total boundedness, [A1] selects a Cauchy subsequence; [F5] makes it convergent in $L^{p} (0, T; H)$ . This proves relative compactness. If the $L^{\infty} (H)$ bound is $M$ , pairwise differences have $L^{\infty} (H)$ norm at most $2 M$ . For $r < p$ , finite-interval Hölder gives convergence in $L^{r} (H)$ ; for $r \geq p$ , $∥ u_{j} - u_{k} ∥_{L^{r} (H)}^{r} \leq (2 M)^{r - p} ∥ u_{j} - u_{k} ∥_{L^{p} (H)}^{p} .$ Completeness in each $L^{r} (H)$ then gives convergence for every finite $r$ . These limits equal the $L^{p} (H)$ limit since both convergences imply convergence in measure, whose limit is unique up to almost-everywhere equality.

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Weak compactness of bounded space-time Sobolev gradients

Statement

Assume the Axiom of Choice. Let $Ω \subseteq R^{d}$ be bounded and open, $d \geq 1$ , let $1 < p < \infty$ , and let $0 < T < \infty$ . Put $V = W_{0}^{1, p} (Ω; R)$ with gradient norm $∥ v ∥_{V} = ∥ \nabla v ∥_{L^{p}}$ , and suppose $V$ has a continuous inclusion in a real Hilbert space $H$ . If $u_{j}$ is bounded in $L^{p} (0, T; V)$ and its image in $L^{p} (0, T; H)$ converges strongly to $u$ , then a subsequence converges weakly in $L^{p} (0, T; V)$ to some $U$ , the image of $U$ equals $u$ , and $\nabla u_{j} ⇀ \nabla U$ in $L^{p} ((0, T) \times Ω; R^{d})$ .

Facts & Assumptions

Given: The Axiom of Choice; bounded open $Ω$ , $d \geq 1$ , $1 < p < \infty$ , $0 < T < \infty$ ; the continuous inclusion $V ↪ H$ ; and a bounded sequence in $L^{p} (0, T; V)$ converging strongly in $L^{p} (0, T; H)$ .

[F1]

$W^{1, p} (Ω)$ is Banach, $W_{0}^{1, p} (Ω)$ is its closed zero-boundary subspace, and Poincaré makes the gradient norm equivalent to the inherited Sobolev norm (Integer-order Sobolev spaces are Banach, Zero-boundary Sobolev space as a norm closure, A closed subspace of a Banach space is Banach, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F2]

For $1 < p < \infty$ , real scalar $L^{p}$ over any measure space is reflexive under Countable Choice (Reflexivity of Lp for one less p less infinity).

[F3]

[F4]

Under Hahn–Banach, every closed linear subspace of a reflexive Banach space is reflexive (Closed subspaces of reflexive spaces are reflexive).

[F5]

The space $L^{p} (0, T; V)$ is complete when $V$ is Banach (Bochner Lp is complete for a Banach value space); the inclusion $V \to H$ induces a bounded linear map $L^{p} (0, T; V) \to L^{p} (0, T; H)$ by the same continuous embedding (Banach evolution triples and weak time derivatives).

[F6]

For $X = L^{p} (Ω; R^{d})$ , finite-valued simple functions are dense in Bochner $L^{p} (0, T; X)$ . To see this, let $s_{n} \to g$ almost everywhere be the simple sequence from strong measurability. For each integer $m \geq 1$ , put $A_{m} = {t : ∥ g (t) ∥_{X} \leq m}$ ; dominated convergence for indicators lets us choose $n_{m}$ so $∣ {t \in A_{m} : ∥ s_{n_{m}} (t) - g (t) ∥_{X} > 1 / m} ∣ \leq m^{- p - 1}$ . Restrict $s_{n_{m}}$ to the complementary good subset of $A_{m}$ and set it to zero elsewhere. This is an integrable simple function. Its $p$ th-power error is at most $T m^{- p} + m^{- 1} + \int_{{∥ g ∥_{X} > m}} ∥ g ∥_{X}^{p}$ , which tends to zero. For a simple $g (t) = \sum_{ℓ = 1}^{q} 1_{E_{ℓ}} (t) g_{ℓ}$ with disjoint measurable $E_{ℓ}$ , choose measurable representatives of each $g_{ℓ} \in L^{p} (Ω; R^{d})$ and set $G (t, x) = \sum_{ℓ = 1}^{q} 1_{E_{ℓ}} (t) g_{ℓ} (x)$ . Each summand is product measurable: $g_{ℓ} \circ π_{Ω}$ is measurable since the inverse image of a measurable set is $(0, T)$ times that set, and $1_{E_{ℓ}} \circ π_{T}$ is measurable by the same rectangle argument; their product is measurable coordinatewise. The spaces are finite-measure Lebesgue spaces because $T < \infty$ and $Ω$ is bounded (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure). Tonelli then gives $∥ G ∥_{L^{p} ((0, T) \times Ω)}^{p} = \sum_{ℓ} ∣ E_{ℓ} ∣ ∥ g_{ℓ} ∥_{L^{p} (Ω)}^{p}$ . Thus this correspondence extends by completeness to an isometric map into $L^{p} ((0, T) \times Ω; R^{d})$ (Banach evolution triples and weak time derivatives, Strongly measurable Banach-valued function, Bochner-integrable function, The product measure of two sigma-finite measure spaces, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Dominated convergence, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[A1]

Countable Choice is supplied by Dependent Choice. The weak subsequence argument uses the ultrafilter lemma, Dependent Choice and Hahn–Banach; all are supplied from the assumed Axiom of Choice (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, Hahn-Banach dominated extension theorem for real vector spaces, The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The real dominated-extension principle as an additional hypothesis over ZF, Dependent choice implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). The countable selections in [F6] use this same Countable Choice branch.

Proof

1.1F1givenalgebra

If $Ω = \emptyset$ , the closure definition in [F1] gives $V = {0}$ , so every $u_{j}$ and its $H$ image is zero. The assumed strong $L^{p} (0, T; H)$ limit is therefore $u = 0$ even when $H$ is nonzero; taking $U = 0$ proves all conclusions.

1.2A1F2F3given

Assume $Ω \neq \emptyset$ , put $Q = (0, T) \times Ω$ , and give $Y_{0} = (L^{p} (Q; R))^{d}$ its maximum product norm. Each scalar coordinate space is reflexive Banach by [F2], so $Y_{0}$ is Banach: a Cauchy sequence converges in each coordinate and therefore in the maximum norm. A bounded sequence in $Y_{0}$ has bounded coordinates; apply [F3] successively to the $d$ coordinates to obtain one subsequence weakly converging in every coordinate. A bounded functional on $Y_{0}$ is the sum of its restrictions to the coordinate embeddings, so this is weak convergence in $Y_{0}$ . The reverse implication in [F3] makes $Y_{0}$ reflexive. The Euclidean-vector norm of $Y = L^{p} (Q; R^{d})$ and the $Y_{0}$ product norm are equivalent because the Euclidean norm and the finite-dimensional $ℓ^{p}$ norm on $R^{d}$ are equivalent. Equivalent norms have the same continuous dual and weak convergence, so $Y$ is reflexive as well.

1.3F1givenconstruct

The gradient map $Γ : V \to L^{p} (Ω; R^{d})$ , $Γ v = \nabla v$ , is an isometry for the chosen norm and has closed range: if $\nabla v_{n}$ converges in $L^{p}$ , then $(v_{n})$ is Cauchy in $V$ ; [F1] gives a limit in $V$ whose gradient is that limit.

1.4A1F2F4F5F6givenconstruct

For a finite-valued simple $U (t) = \sum_{ℓ = 1}^{q} 1_{E_{ℓ}} (t) v_{ℓ}$ with $v_{ℓ} \in V$ and disjoint measurable $E_{ℓ}$ , apply [F6] to $Γ U (t) = \sum_{ℓ} 1_{E_{ℓ}} (t) \nabla v_{ℓ}$ . It supplies the space-time representative and equality of the Bochner and product $L^{p}$ norms. Simple-function density from [F6] and completeness of $Y$ from [F2] extend this map to an isometry $Γ_{p} : L^{p} (0, T; V) \to Y$ . Its range $S$ is closed: if $Γ_{p} U_{n}$ converges in $Y$ , the isometry makes $U_{n}$ Cauchy in $L^{p} (0, T; V)$ ; completeness [F5] gives $U_{n} \to U$ , and then $Γ_{p} U_{n} \to Γ_{p} U$ . Thus [F4] makes $S$ a reflexive Banach space.

2.1A1F3step 1.4

The sequence $Γ_{p} u_{j}$ is bounded in $S$ . Reflexivity of $S$ and [F3] give a weakly convergent subsequence with limit $G \in S$ . Since $G = Γ_{p} U$ for some $U \in L^{p} (0, T; V)$ and $Γ_{p}$ is an isometry, the same subsequence $u_{j}$ converges weakly to $U$ in $L^{p} (0, T; V)$ ; applying any bounded functional after $Γ_{p}^{- 1}$ verifies this assertion.

3.1F5F6step 2.1givenalgebra

The bounded inclusion in [F5] sends this weak convergence to weak convergence of the $H$ -valued images to the image of $U$ . By hypothesis those same images converge strongly, hence weakly, to $u$ ; Hahn–Banach separates distinct points, so $u$ is the image of $U$ . Weak convergence in $S$ also implies weak convergence of gradients in $Y$ , by restriction of $Y^{*}$ functionals to $S$ .

4.1step 2.1step 3.1∎

The subsequence therefore has the asserted weak $L^{p} (V)$ limit, that limit has $H$ image $u$ , and its space-time gradient is the weak limit $G = \nabla U$ . This proves all conclusions.

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Piecewise-constant discrete Aubin–Lions compactness

Statement

Assume the Axiom of Choice.

Let $V ⋐ H ↪ V^{*}$ be a Banach evolution triple. For each $k$ , let $0 = t_{0}^{k} < \dots < t_{N_{k}}^{k} = T$ be a partition with maximum step $δ_{k} \to 0$ , let $U_{k}^{n} \in V$ , and define ${\overline{U}}_{k} (t) = U_{k}^{n}$ on $I_{n}^{k} = (t_{n - 1}^{k}, t_{n}^{k}]$ and $D_{k} (t) = (U_{k}^{n} - U_{k}^{n - 1}) / τ_{n}^{k}$ there. Suppose, for some $1 < p < \infty$ and $1 < q \leq \infty$ , $\sup_{k} ∥ {\overline{U}}_{k} ∥_{L^{p} (0, T; V)} + \sup_{k} ∥ D_{k} ∥_{L^{q} (0, T; V^{*})} < \infty .$ Then the time translates satisfy, on their common intervals, $∥ {\overline{U}}_{k} (\cdot + s) - {\overline{U}}_{k} ∥_{L^{q} (V^{*})} \leq (s + 2 δ_{k}) ∥ D_{k} ∥_{L^{q} (V^{*})},$ and $({\overline{U}}_{k})$ is relatively compact in $L^{p} (0, T; H)$ . If also $\sup_{k, n} ∥ U_{k}^{n} ∥_{H} < \infty$ , every convergent subsequence converges in every $L^{r} (0, T; H)$ with $1 \leq r < \infty$ .

Facts & Assumptions

Given: The Axiom of Choice, the compact and continuous embeddings, the partitions and node values, and the two uniform Bochner bounds in the statement.

[F1]

$V ⋐ H$ means bounded subsets of $V$ have compact closure in $H$ ; $H ↪ V^{*}$ is the canonical evolution embedding (Compactly embedded normed spaces, Banach evolution triples and weak time derivatives).

[F2]

Hölder and Minkowski inequalities give $∥ \int_{a}^{b} g ∥ \leq \int_{a}^{b} ∥ g ∥$ and $∥ g + h ∥_{L^{q}} \leq ∥ g ∥_{L^{q}} + ∥ h ∥_{L^{q}}$ (Holder's inequality for integrals, including the endpoint cases, Minkowski's inequality for integrals, including $p = \infty$ ).

[F3]

The Bochner space $L^{p} (0, T; H)$ is complete (Bochner Lp is complete for a Banach value space).

[A1]

The Axiom of Choice supplies Countable and Dependent Choice for sequential compactness and subsequence extraction (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

1.1F1F2given

View each node in $V^{*}$ through the canonical embeddings $V \to H \to V^{*}$ , and let ${\hat{U}}_{k}$ be their continuous affine interpolation. On $I_{n}^{k}$ its derivative equals $D_{k}$ . For $0 \leq s < T$ and $0 < t < T - s$ , the fundamental theorem for the affine function gives ${\hat{U}}_{k} (t + s) - {\hat{U}}_{k} (t) = \int_{t}^{t + s} D_{k} (r) d r$ . Minkowski [F2] and translation invariance of the time integral yield $∥ {\hat{U}}_{k} (\cdot + s) - {\hat{U}}_{k} ∥_{L^{q} (0, T - s; V^{*})} \leq s ∥ D_{k} ∥_{L^{q} (V^{*})}$ .

2.1F2step 1.1algebra

If $t \in I_{n}^{k}$ , write $t = t_{n - 1}^{k} + θ τ_{n}^{k}$ with $0 < θ \leq 1$ . Then ${\overline{U}}_{k} (t) - {\hat{U}}_{k} (t) = (1 - θ) (U_{k}^{n} - U_{k}^{n - 1})$ . Consequently, for $1 < q < \infty$ , $∥ {\overline{U}}_{k} - {\hat{U}}_{k} ∥_{L^{q} (V^{*})}^{q} \leq \sum_{n} τ_{n}^{k} ∥ U_{k}^{n} - U_{k}^{n - 1} ∥_{V^{*}}^{q} \leq δ_{k}^{q} ∥ D_{k} ∥_{L^{q} (V^{*})}^{q} .$ For $q = \infty$ , the same pointwise identity gives $∥ {\overline{U}}_{k} (t) - {\hat{U}}_{k} (t) ∥_{V^{*}} \leq δ_{k} ∥ D_{k} ∥_{L^{\infty} (V^{*})}$ and hence $∥ {\overline{U}}_{k} - {\hat{U}}_{k} ∥_{L^{\infty} (V^{*})} \leq δ_{k} ∥ D_{k} ∥_{L^{\infty} (V^{*})}$ . Combining the corresponding estimate at $t$ and $t + s$ with step 1.1 proves the stated $(s + 2 δ_{k})$ translate bound in both cases.

3.1F1F2step 2.1choose

Fix $ε > 0$ . By [F1], the image in $H$ of the $V$ unit ball has compact closure. Choose finitely many centers from that image forming an $ε$ -net, let $E \subset i (V)$ be their span, and let $P_{E}$ be the $H$ -orthogonal projection. Its best-approximation property gives $∥ (I - P_{E}) i v ∥_{H} \leq ε ∥ v ∥_{V}$ for every $v \in V$ . Choose an $H$ -orthonormal basis $e_{j} = i v_{j}$ of $E$ . The scalar coefficients $a_{k, j}^{n} = (U_{k}^{n}, e_{j})_{H}$ have weighted $L^{p}$ bounds from the $L^{p} (V)$ bound, and their total variation satisfies $\sum_{n} ∣ a_{k, j}^{n} - a_{k, j}^{n - 1} ∣ \leq ∥ v_{j} ∥_{V} \sum_{n} τ_{n}^{k} ∥ D_{k} ∥_{V^{*}} \leq T^{1 - 1 / q} ∥ v_{j} ∥_{V} ∥ D_{k} ∥_{L^{q} (V^{*})} .$ At least one node value is no larger than its weighted $L^{p}$ average, so the total-variation bound gives a uniform bound for every nodal coefficient.

4.1F2step 3.1choose

The affine scalar interpolants of these finitely many coefficients are uniformly bounded; their derivatives are the pairings of $D_{k}$ with $v_{j}$ , so their $L^{q}$ norms are uniformly bounded. Hölder gives a common modulus $C ∣ t - s ∣^{1 - 1 / q}$ (or $C ∣ t - s ∣$ for $q = \infty$ ). Choose a finite time grid on which the modulus between adjacent points is at most $ε$ , and finite $ε$ -nets for the bounded coefficient vectors at each grid point. The piecewise-linear interpolants of the rounded grid values form a finite family within $2 ε$ uniformly of every coefficient vector, so the projected affine family is totally bounded in $C ([0, T]; E)$ . The maximum nodal jump is at most $δ_{k}^{1 - 1 / q} ∥ D_{k} ∥_{L^{q} (V^{*})} ∥ v_{j} ∥_{V}$ , so the projected piecewise-constant family differs from it uniformly by a quantity tending to zero. Thus ${P_{E} {\overline{U}}_{k}}$ is relatively compact in $L^{p} (H)$ .

5.1A1F1F3step 3.1step 4.1

The projection remainder satisfies $∥ {\overline{U}}_{k} - P_{E} {\overline{U}}_{k} ∥_{L^{p} (H)} \leq ε ∥ {\overline{U}}_{k} ∥_{L^{p} (V)}$ . Since $ε$ is arbitrary, steps 3.1 and 4.1 show the full family is totally bounded in $L^{p} (H)$ ; [A1] extracts a Cauchy subsequence, which converges by [F3]. This proves relative compactness.

6.1F2F3step 5.1algebra∎

If the nodal values are uniformly bounded by $M$ in $H$ , then $∥ {\overline{U}}_{k} - {\overline{U}}_{ℓ} ∥_{L^{\infty} (H)} \leq 2 M$ . For $r < p$ , finite-measure Hölder upgrades the strong $L^{p} (H)$ convergence to $L^{r} (H)$ . For $r \geq p$ , $∥ {\overline{U}}_{k} - {\overline{U}}_{ℓ} ∥_{L^{r} (H)}^{r} \leq (2 M)^{r - p} ∥ {\overline{U}}_{k} - {\overline{U}}_{ℓ} ∥_{L^{p} (H)}^{p}$ , so the same subsequence is Cauchy in every finite $L^{r} (H)$ and converges there by [F3]. These limits equal the strong $L^{p} (H)$ limit since both convergences imply convergence in measure, whose limit is unique up to almost-everywhere equality.

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Boundary-preserving quasi-interpolation is stable and strongly approximating

Statement

Assume Countable Choice. Use either mesh branch and the nodal space defined in Quasi-uniform finite-element mass inverses have summable graph decay. For $d \geq 2$ , assume $Ω$ is a bounded polyhedral open set whose boundary is locally a graph of a Lipschitz function with $Ω$ on one side; for $d = 1$ , use a bounded nonempty interval or a finite nonempty union of bounded intervals with pairwise disjoint closures. Let $1 \leq p < \infty$ . There is a linear map

Q_{h} : W_{0}^{1, p} (Ω) ⟶ V_{h}

with the following construction. Fix a nonnegative $ρ \in C_{c}^{\infty} (B (0, 1))$ with $\int ρ = 1$ , let $E_{0}$ be extension by zero, and set $S_{h} v = ρ_{h} * E_{0} v$ with $ρ_{h} (x) = h^{- d} ρ (x / h)$ . The degree- $k$ coefficient of $Q_{h} v$ at every interior node $z$ is $S_{h} v (z)$ ; every boundary-node coefficient is zero. For all sufficiently fine meshes, $∥ v - Q_{h} v ∥_{L^{p} (Ω)} \leq C h ∥ \nabla v ∥_{L^{p} (Ω)}, ∥ \nabla Q_{h} v ∥_{L^{p} (Ω)} \leq C ∥ \nabla v ∥_{L^{p} (Ω)} .$ Consequently $Q_{h} v \to v$ in $W^{1, p} (Ω)$ for each $v \in W_{0}^{1, p} (Ω)$ . Constants are uniform over the fixed mesh family and may depend on $d, k, p, Ω, θ, σ$ and $ρ$ . The operator $Q_{h}$ is not asserted to be a projector.

Facts & Assumptions

Given: Countable Choice, one mesh branch of Quasi-uniform finite-element mass inverses have summable graph decay, its nodal space $V_{h}$ , a fixed $1 \leq p < \infty$ , and $v \in W_{0}^{1, p} (Ω)$ .

[F1]

On each cell, the Lagrange coefficient estimates scale like $h_{K}^{d / p}$ in $L^{p}$ and like $h_{K}^{d / p - 1}$ for gradients; the local basis reproduces constants and affine polynomials. Globally the coefficient estimates scale like $h^{d / p}$ ; graph balls have polynomial growth and the mesh families have the stated $θ, σ$ bounds (Quasi-uniform finite-element mass inverses have summable graph decay).

[F2]

For every open $Ω$ , zero extension maps $W_{0}^{1, p} (Ω)$ into $W^{1, p} (R^{d})$ and preserves the function and weak-gradient norms (Zero extension of W_0^{1,p} has no boundary derivative).

[F3]

On a matching simplex mesh, a continuous piecewise polynomial with zero polynomial trace on every boundary facet belongs to $W_{0}^{1, p}$ for finite $p$ (Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare). The one-dimensional interval case is proved locally below.

[F4]

A unit-mass smooth bump generates a smooth convolution and derivatives pass under the integral; its mollifier family is an $L^{1}$ approximate identity, and convolution by it converges in every finite $L^{p}$ (Convolution with a mollifier is smooth, and derivatives pass under the integral sign, Interior mollification commutes with weak derivatives, The mollifier family generated by a unit-mass smooth bump, A unit-mass smooth bump generates an $L^{1}$ approximate identity, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ ).

[F5]

Hölder's inequality bounds convolution evaluation against an $L^{p}$ function by the $L^{p^{'}}$ norm of the kernel (Holder's inequality for integrals, including the endpoint cases).

[F6]

A differentiable curve on a compact interval satisfies its integral fundamental theorem, and compactly supported $W^{1, p} (R^{d})$ functions can be approximated by compactly supported smooth functions (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F7]

The support of a convolution is contained in the closure of the sum of the supports (The support of a convolution lies in the closure of the support sumset).

[F8]

$W_{0}^{1, p}$ is the norm closure of $C_{c}^{\infty} (Ω)$ and its norm uses the function and first weak derivatives (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[F9]

The smooth bump lemma supplies a nonnegative compactly supported $ρ_{0}$ equal to one on an inner ball; its integral is finite and positive because its support is bounded and it is one on a ball containing a box of positive volume. Thus $ρ = ρ_{0} / \int ρ_{0}$ is an admissible unit-mass bump (A smooth bump between concentric Euclidean balls, A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[A1]

Countable Choice is assumed in the zero-extension, mollification, Sobolev-closure, and approximation interfaces cited above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F1F3F6F8givenconstruct

The coefficients define a linear map into the algebraic conforming piecewise-polynomial space: shared-face nodes coincide, so the face polynomials have identical nodal data and hence identical traces; all boundary coefficients are zero, so each boundary-facet polynomial vanishes. For $d \geq 2$ , [F3] therefore gives $V_{h} \subset W_{0}^{1, p} (Ω)$ . For $d = 1$ , write a mesh function as $w$ , let $M$ bound its finitely many piecewise derivatives, and extend it by zero. It is continuous at every mesh node and at every component endpoint. Integrating the derivative on each of the finitely many subintervals between two points using the fundamental theorem in [F6] shows $∣ w (x) - w (y) ∣ \leq M ∣ x - y ∣$ on $R$ ; integration by parts on those subintervals shows that the zero extension has the piecewise derivative as its weak derivative. Thus it is compactly supported in $W^{1, p} (R)$ .

1.2A1F6givenalgebra

Let $u = E_{0} v$ . For a cell $K$ , choose a cube $C_{K}$ of side $R h$ centered at an interior point of $K$ , where the fixed $R$ is large enough that $K$ , every radius- $h$ ball centered at one of its nodes, and every radius- $2 h$ ball centered at a point of $K$ lie in $C_{K}$ . The cubes have bounded overlap: if $x \in C_{K}$ , then $K \subset B (x, C_{R} h)$ ; the cells have disjoint interiors and volume at least $c h^{d}$ , so only uniformly many can meet that ball. For smooth $u$ on a cube $C$ , the segment fundamental theorem, Hölder, Jensen's inequality, and a change of variables on the two ranges $t \leq 1 / 2$ and $t \geq 1 / 2$ give $∥ u - u_{C} ∥_{L^{p} (C)} \leq C_{d} diam (C) ∥ \nabla u ∥_{L^{p} (C)}, u_{C} = ∣ C ∣^{- 1} \int_{C} u .$ Indeed, Jensen's inequality and $u (x) - u (y) = \int_{0}^{1} \nabla u (t x + (1 - t) y) \cdot (x - y) d t$ give $∥ u - u_{C} ∥_{L^{p} (C)}^{p} \leq \frac{diam (C)^{p}}{∣ C ∣} \int_{C} \int_{C} \int_{0}^{1} ∣ \nabla u (t x + (1 - t) y) ∣^{p} d t d y d x .$ For $t \geq 1 / 2$ change the $x$ variable, and for $t \leq 1 / 2$ change the $y$ variable; both inverse Jacobians are at most $2^{d}$ and the images stay in $C$ . This proves the displayed Poincaré estimate. Since $u = E_{0} v$ is compactly supported in $W^{1, p} (R^{d})$ , [F6] approximates it by smooth functions in that norm, so the same cube estimate holds for $u$ .

2.1A1F4F6F7F8F9step 1.1givenalgebra

In the one-dimensional branch, for $x$ in a component interval choose the nearer endpoint; integrating the derivative from that endpoint by [F6] gives $∣ w (x) ∣ \leq M dist (x, R ∖ Ω)$ . Because there are finitely many distinct endpoints, choose $ε$ smaller than one quarter of their minimum separation. Translate and rescale the bump from [F9] at each endpoint to obtain $χ_{a, ε}$ equal to one within $ε$ of $a$ and zero outside distance $2 ε$ . The endpoint supports are disjoint; the product $η_{ε} = \prod_{a \in \partial Ω} (1 - χ_{a, ε})$ is smooth, vanishes within distance $ε$ of each endpoint, equals one at distance at least $2 ε$ , and satisfies $∣ η_{ε}^{'} ∣ \leq C / ε$ . Thus $w - η_{ε} w$ and its weak derivative are supported in a boundary layer of measure $O_{Ω} (ε)$ and bounded there by $C_{Ω} M$ . On each partition interval $η_{ε} w$ is smooth times polynomial, is continuous at nodes, and vanishes near component endpoints; integration by parts on the finitely many intervals cancels all interior endpoint terms. Its zero extension therefore belongs to $W^{1, p} (R)$ with derivative $η_{ε}^{'} w + η_{ε} w^{'}$ on the pieces. Hence $η_{ε} w \to w$ in $W^{1, p} (R)$ . Each cutoff product has compact support a positive distance from $R ∖ Ω$ ; convolving it with a sufficiently small mollifier gives a function in $C_{c}^{\infty} (Ω)$ by [F4] and [F7], and the convolution converges in $W^{1, p}$ by [F4]. This proves $w \in W_{0}^{1, p} (Ω)$ by [F8] and completes the range assertion in $d = 1$ .

2.2A1step 1.2givenalgebra

If $K$ has no boundary node, put $c_{K} = u_{C_{K}}$ . If it has a boundary node, put $c_{K} = 0$ . In $d \geq 2$ , compactness of the boundary gives a finite subcover of its Lipschitz graph charts; taking the largest chart Lipschitz constant and the smallest chart radius yields uniform $η, r_{0} > 0$ such that each boundary point $z$ and $0 < r \leq r_{0}$ have an exterior cone sector in $B (z, r)$ of measure at least $η r^{d}$ : in a chart $x_{d} > g (x^{'})$ with Lipschitz constant $L$ , the cone $x_{d} < g (z^{'}) - L ∣ x^{'} - z^{'} ∣$ is exterior, and a fixed smaller cone has that volume. For sufficiently small $h$ , a boundary node $z$ of $K$ therefore gives $∣ C_{K} ∖ Ω ∣ \geq η^{'} ∣ C_{K} ∣$ . In $d = 1$ , at a boundary endpoint the exterior interval inside $C_{K}$ has a fixed positive fraction of its length; for finitely many separated components take $h$ smaller than their minimum gap divided by $2 R$ . In the boundary case $u = 0$ on $C_{K} ∖ Ω$ , so $∣ u_{C_{K}} ∣ ∣ C_{K} ∖ Ω ∣^{1 / p} \leq ∥ u - u_{C_{K}} ∥_{L^{p} (C_{K} ∖ Ω)} .$ The cube estimate of step 1.2 then gives, in either case, $∥ u - c_{K} ∥_{L^{p} (C_{K})} \leq C h ∥ \nabla u ∥_{L^{p} (C_{K})} .$

2.3F1F4step 1.1givenalgebra

Fix $ψ \in C_{c}^{\infty} (Ω)$ and let $δ = dist (supp ψ, \partial Ω) > 0$ . For $h < δ / 2$ , $S_{h} ψ$ is zero on every cell with a boundary node: every point of such a cell is within distance $h$ of the boundary, while the convolution samples only a radius- $h$ ball. Thus $Q_{h} ψ = 0 = S_{h} ψ$ on those cells. Every other cell has only interior nodes, so $Q_{h} ψ$ is the ordinary nodal interpolant of $S_{h} ψ$ there. Since $k \geq 1$ , this interpolant reproduces affine polynomials. Taylor's formula about a cell vertex and the local coefficient estimates in [F1] give $∥ Q_{h} ψ - S_{h} ψ ∥_{L^{p} (K)} \leq C h^{2} ∣ K ∣^{1 / p} ∥ D^{2} ψ ∥_{\infty}, ∥ \nabla (Q_{h} ψ - S_{h} ψ) ∥_{L^{p} (K)} \leq C h ∣ K ∣^{1 / p} ∥ D^{2} ψ ∥_{\infty} .$ The nonnegative unit-mass kernel gives $S_{h} ψ \to ψ$ in $W^{1, p} (Ω)$ and $∥ D^{2} S_{h} ψ ∥_{\infty} \leq ∥ D^{2} ψ ∥_{\infty}$ . Summing cell bounds proves $Q_{h} ψ \to ψ$ in $W^{1, p}$ .

3.1F1F5step 2.2givenalgebra

For each interior node $z$ of $K$ , Hölder [F5] gives $∣ S_{h} v (z) - c_{K} ∣ \leq ∥ ρ_{h} ∥_{L^{p^{'}} (R^{d})} ∥ u - c_{K} ∥_{L^{p} (B (z, h))} \leq C h^{- d / p} ∥ u - c_{K} ∥_{L^{p} (C_{K})} .$ At a boundary node both the coefficient and $c_{K}$ are zero in the boundary case. If $K$ has no boundary node, every local node is interior and the local nodal basis reproduces the constant $c_{K}$ . Applying the local coefficient estimate of [F1] to the coefficients of $Q_{h} v - c_{K}$ , and using step 2.2, gives $∥ Q_{h} v - c_{K} ∥_{L^{p} (K)} \leq C h ∥ \nabla u ∥_{L^{p} (C_{K})}, ∥ \nabla Q_{h} v ∥_{L^{p} (K)} \leq C ∥ \nabla u ∥_{L^{p} (C_{K})} .$ The boundary-node case uses $c_{K} = 0$ ; the no-boundary-node case uses $\sum_{z \in K} ϕ_{z} = 1$ .

4.1F1F2step 1.2step 3.1givenalgebra

Since $v = u$ on $K$ , step 2.2 and the first estimate of step 3.1 imply $∥ v - Q_{h} v ∥_{L^{p} (K)} \leq C h ∥ \nabla u ∥_{L^{p} (C_{K})} .$ Raise this and the gradient estimate of step 3.1 to the $p$ th power, sum over cells, and use the bounded overlap from step 1.2 and the norm preservation in [F2]. This proves both global estimates in the Statement. Also $∥ Q_{h} w ∥_{W^{1, p}} \leq C ∥ w ∥_{W^{1, p}}$ for every $w \in W_{0}^{1, p}$ and all sufficiently small $h$ , by the gradient bound and $∥ Q_{h} w ∥_{p} \leq ∥ w ∥_{p} + C h ∥ \nabla w ∥_{p}$ .

5.1F8step 4.1step 2.3givenalgebra∎

For arbitrary $v \in W_{0}^{1, p} (Ω)$ and $ε > 0$ , [F8] provides $ψ \in C_{c}^{\infty} (Ω)$ with $∥ v - ψ ∥_{W^{1, p}} < ε$ . The uniform bound in step 4.1 and the smooth convergence in step 2.3 imply ${lim sup}_{h \to 0} ∥ Q_{h} v - v ∥_{W^{1, p}} \leq (C + 1) ∥ v - ψ ∥_{W^{1, p}} < (C + 1) ε .$ Letting $ε ↓ 0$ proves strong convergence for every $v \in W_{0}^{1, p} (Ω)$ .

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Minty identification for a monotone Carathéodory flux

Statement

Assume the Axiom of Choice.

Let $Q = (0, T) \times Ω$ have finite measure, $1 < p < \infty$ , and let $a (x, ξ)$ be a Carathéodory flux with the monotonicity and $p$ -growth hypotheses of Coercive monotone p-growth diffusion fluxes, independent of time. Suppose $ξ_{n} ⇀ ξ$ in $L^{p} (Q; R^{d})$ , $q_{n} = a (x, ξ_{n}) ⇀ q$ in $L^{p^{'}} (Q; R^{d})$ , and ${lim sup}_{n \to \infty} \int_{Q} q_{n} \cdot ξ_{n} \leq \int_{Q} q \cdot ξ .$ Then $q (x, t) = a (x, ξ (x, t))$ for almost every $(t, x) \in Q$ .

Facts & Assumptions

Given: The Axiom of Choice, the finite-measure cylinder, $1 < p < \infty$ , the stated flux hypotheses, weak convergences, and the limsup inequality.

[F1]

The Carathéodory composition lemma makes $a (x, η (t, x))$ measurable, and the growth bound gives $a (\cdot, η) \in L^{p^{'}} (Q; R^{d})$ for every $η \in L^{p} (Q; R^{d})$ (A Caratheodory integrand composed with measurable functions is measurable, Coercive monotone p-growth diffusion fluxes).

[F2]

The flux is pointwise monotone outside one null set: $(a (x, ξ) - a (x, η)) \cdot (ξ - η) \geq 0$ (Coercive monotone p-growth diffusion fluxes).

[F3]

Hölder makes every product of an $L^{p^{'}}$ factor and an $L^{p}$ factor integrable (Holder's inequality for integrals, including the endpoint cases).

[F4]

Dominated convergence passes limits under an integrable pointwise majorant (Dominated convergence).

Proof

1.1F1F2F3givenalgebra

Fix arbitrary $η \in L^{p} (Q; R^{d})$ . By [F1]–[F3], monotonicity gives the finite nonnegative integrals $0 \leq \int_{Q} (q_{n} - a (x, η)) \cdot (ξ_{n} - η) .$ Expanding them and using the two weak convergences for the three terms linear in a weakly convergent factor, the limsup hypothesis yields $0 \leq \int_{Q} (q - a (x, η)) \cdot (ξ - η) .$ Indeed the expanded nonnegative sequence has limsup at most the displayed right-hand side, so that right-hand side cannot be negative.

2.1F1F3F4step 1.1givenalgebra

Let $ζ \in L^{p} (Q; R^{d})$ and $0 < ε \leq 1$ . Substitute $η = ξ + ε ζ$ and $η = ξ - ε ζ$ into step 1.1. After dividing by $ε$ , the two inequalities give $\int_{Q} (q - a (x, ξ + ε ζ)) \cdot ζ \leq 0, \int_{Q} (q - a (x, ξ - ε ζ)) \cdot ζ \geq 0.$ The fixed term $q \cdot ζ$ is integrable by [F3]. For either sign, the flux term satisfies $∣ a (x, ξ \pm ε ζ) ∣ ∣ ζ ∣ \leq C (g (x) + ∣ ξ ∣^{p - 1} + ∣ ζ ∣^{p - 1}) ∣ ζ ∣,$ which is integrable by [F3]; here $g (x)$ is viewed as constant in time and $Q$ has finite time length. Carathéodory continuity gives pointwise $a (x, ξ \pm ε ζ) \to a (x, ξ)$ as $ε ↓ 0$ . Dominated convergence [F4] on the flux term in both inequalities proves $\int_{Q} (q - a (x, ξ)) \cdot ζ = 0$ for every $ζ \in L^{p} (Q; R^{d})$ .

3.1F3step 2.1algebra∎

Put $w = q - a (x, ξ) \in L^{p^{'}} (Q; R^{d})$ and choose $ζ = ∣ w ∣^{p^{'} - 2} w$ , defined as zero where $w = 0$ . Since $(p^{'} - 1) p = p^{'}$ , this $ζ$ belongs to $L^{p}$ , and step 2.1 gives $0 = \int_{Q} w \cdot ζ = \int_{Q} ∣ w ∣^{p^{'}}$ . Thus $w = 0$ almost everywhere, which is the claimed flux identification.

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Quantitative residual error under strong monotonicity

Statement

Assume the Axiom of Choice.

Let $(V, H, J)$ be the evolution triple of Banach evolution triples and weak time derivatives, $1 0$ and exponent $p$ : $⟨ A (x) - A (y), x - y ⟩ \geq μ ∥ x - y ∥_{V}^{p} (x, y \in V) .$ Suppose $u, v \in L^{p} (0, T; V)$ have derivatives in $L^{p^{'}} (0, T; V^{*})$ and residual difference $R = (u^{'} + A (u)) - (v^{'} + A (v)) \in L^{p^{'}} (0, T; V^{*})$ . Then for every $t \in [0, T]$ , $∥ u (t) - v (t) ∥_{H}^{2} + μ \int_{0}^{t} ∥ u - v ∥_{V}^{p} \leq ∥ u (0) - v (0) ∥_{H}^{2} + 2 C_{μ, p} ∥ R ∥_{L^{p^{'}} (0, t; V^{*})}^{p^{'}},$ where both functions use their continuous $H$ representatives and $C_{μ, p} = \frac{1}{p^{'}} (2 / (μ p))^{1 / (p - 1)}$ . In particular, equal initial data and zero residual imply $u = v$ .

Facts & Assumptions

Given: The Axiom of Choice, the evolution triple, $1 < p < \infty$ , strong monotonicity with constant $μ$ , two functions in the stated evolution space, and their residual difference $R$ .

[F1]

Functions in $L^{p} (V)$ with derivative in $L^{p^{'}} (V^{*})$ have continuous $H$ representatives and satisfy the integrated Hilbert norm chain rule (The Banach evolution-space energy identity and H trace).

[F2]

Hölder's inequality applies to the dual pairing of $L^{p^{'}} (V^{*})$ and $L^{p} (V)$ (Holder's inequality for integrals, including the endpoint cases).

[F3]

For conjugate exponents $p, p^{'} > 1$ , Young's inequality is $a b \leq a^{p} / p + b^{p^{'}} / p^{'}$ for $a, b \geq 0$ (Young's inequality for conjugate real exponents).

[A1]

Strong monotonicity with exponent $p$ and constant $μ$ is the additional hypothesis of this theorem; it is not part of ordinary monotonicity in Coercive monotone p-growth diffusion fluxes.

[A2]

The Axiom of Choice licenses the energy chain rule through its absolutely continuous representative prerequisite (The Axiom of Choice).

Proof

1.1A1A2F1F2givenalgebra

Set $e = u - v$ . Subtracting the residual definitions gives $e^{'} + A (u) - A (v) = R$ in $L^{p^{'}} (0, T; V^{*})$ . By [F1], testing this identity against $e \in L^{p} (0, T; V)$ is integrable by [F2], and for every $t$ , $\frac{1}{2} ∥ e (t) ∥_{H}^{2} - \frac{1}{2} ∥ e (0) ∥_{H}^{2} + \int_{0}^{t} ⟨ A (u) - A (v), e ⟩ = \int_{0}^{t} ⟨ R, e ⟩ .$

2.1A1A2F1F2F3step 1.1givenalgebra∎

Strong monotonicity bounds the left integral below by $μ \int_{0}^{t} ∥ e ∥_{V}^{p}$ . Young's inequality [F3], applied to $(μ p / 2)^{1 / p} a$ and $(μ p / 2)^{- 1 / p} b$ , gives the scaled form $a b \leq \frac{μ}{2} a^{p} + \frac{1}{p^{'}} {(\frac{2}{μ p})}^{1 / (p - 1)} b^{p^{'}}$ gives $∣ ⟨ R, e ⟩ ∣ \leq \frac{μ}{2} ∥ e ∥_{V}^{p} + C_{μ, p} ∥ R ∥_{V^{*}}^{p^{'}}$ . Integrating and absorbing the first term yields $\frac{1}{2} ∥ e (t) ∥_{H}^{2} + \frac{μ}{2} \int_{0}^{t} ∥ e ∥_{V}^{p} \leq \frac{1}{2} ∥ e (0) ∥_{H}^{2} + C_{μ, p} ∥ R ∥_{L^{p^{'}} (0, t; V^{*})}^{p^{'}} .$ Multiplying by two proves the estimate. If $e (0) = 0$ and $R = 0$ , the nonnegative left side vanishes for every $t$ , hence $e (t) = 0$ in $H$ and $e = 0$ in $L^{p} (V)$ .

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L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes

Statement

Assume Countable Choice and fix a degree $k \geq 1$ . Take a sequence of meshes with $h \to 0$ in either branch and use the zero-boundary Lagrange spaces $V_{h}$ from Quasi-uniform finite-element mass inverses have summable graph decay: globally quasi-uniform, shape-regular matching affine simplices on a bounded Lipschitz polyhedral open set in $d \geq 2$ (with the local graph-chart convention of Boundary-preserving quasi-interpolation is stable and strongly approximating), or interval partitions of a nonempty bounded interval (or a finite nonempty union of intervals with disjoint closures) in $d = 1$ . For $f \in L^{1} (Ω)$ define $P_{h} f \in V_{h}$ by the finite mass system

\int_{Ω} (P_{h} f) w_{h} = \int_{Ω} f w_{h} (w_{h} \in V_{h}) .

This is well-defined, agrees on $L^{2} (Ω)$ with the $L^{2}$ -orthogonal projection, and is the identity on $V_{h}$ . There are constants independent of $h$ such that $∥ P_{h} f ∥_{L^{r} (Ω)} \leq C_{r} ∥ f ∥_{L^{r} (Ω)} (1 \leq r \leq \infty) .$ For every $1 < p < \infty$ and $v \in W_{0}^{1, p} (Ω)$ , $∥ \nabla P_{h} v ∥_{L^{p} (Ω)} \leq C_{p} ∥ \nabla v ∥_{L^{p} (Ω)}, P_{h} v ⟶ v in W^{1, p} (Ω) .$ For the PE12 application, additionally assume the Axiom of Choice and $p > \max (1, 2 d / (d + 2))$ . The compact embedding The subcritical W0 one p embedding into L2 makes $V = W_{0}^{1, p} (Ω)$ a dense subspace of $H = L^{2} (Ω)$ , and The zero-boundary W1p space is separable and reflexive supplies the separable reflexive Banach structure. Thus these spaces form the evolution triple of Coercive monotone p-growth diffusion fluxes and Banach evolution triples and weak time derivatives, and the $P_{h}$ above is its $H$ -orthogonal projector. Together with any time partitions whose maximum step tends to zero it satisfies the projector clauses in Conforming FEM families with stable L2 projectors. For every $d \geq 2$ , the uniform Kuhn/Freudenthal meshes of $Ω = (0, 1)^{d}$ , and for $d = 1$ the uniform interval meshes of $Ω = (0, 1)$ , are explicit nonempty instances throughout that full strict $p$ range. Time refinement is independent of spatial refinement.

Facts & Assumptions

Given: Countable Choice; fixed $k \geq 1$ ; one of the two mesh branches; and a finite mesh in its family.

[F1]

The scaled mass matrix has uniform spectrum, inverse decay, absolute row and column sums, and coefficient scaling; the one-dimensional interval branch is part of its Statement (Quasi-uniform finite-element mass inverses have summable graph decay).

[F2]

The zero-boundary quasi-interpolant maps $W_{0}^{1, p}$ into $V_{h}$ , is uniformly gradient-stable, and converges strongly in $W^{1, p}$ for every finite $p$ . Its proof establishes $V_{h} \subset W_{0}^{1, p}$ directly on the interval branch as well as the interval quasi-interpolation estimates (Boundary-preserving quasi-interpolation is stable and strongly approximating).

[F3]

On $d \geq 2$ shape-regular matching simplices, degree- $k$ polynomials satisfy the inverse estimate $∥ \nabla w_{h} ∥_{L^{p} (K)} \leq C h_{K}^{- 1} ∥ w_{h} ∥_{L^{p} (K)}$ (Polynomial inverse and discrete trace inequalities). On an interval, the local upper gradient and lower function coefficient bounds in [F1], after division, give the same estimate.

[F4]

The zero-boundary conforming polynomial space on a matching simplex mesh is contained in $W_{0}^{1, p}$ for finite $p$ (Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare); the interval membership branch is proved in [F2].

[F5]

$C_{c}^{\infty} (Ω)$ is dense in $L^{2} (Ω)$ for every open $Ω$ (Smooth compactly supported functions of an open set are dense in $L^{2}$ ).

[F6]

An admissible stable conforming FEM sequence requires finite-dimensional $V_{k} \subset V$ , their $H$ -orthogonal projectors, uniform $V$ -stability, strong $V$ -approximation, and time partitions whose maximum step tends to zero (Conforming FEM families with stable L2 projectors).

[F7]

On an open set bounded in one direction, Poincaré bounds the $L^{p}$ norm of a zero-boundary function by its gradient norm for finite $p$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F8]

Hölder's inequality bounds each mass right-hand-side functional $f \mapsto \int_{Ω} f ϕ_{i}$ on $L^{p} (Ω)$ by $∥ f ∥_{p} ∥ ϕ_{i} ∥_{p^{'}}$ (Holder's inequality for integrals, including the endpoint cases).

[F9]

In the strict PE12 exponent range, $W_{0}^{1, p} (Ω)$ is densely, continuously, and compactly embedded in $L^{2} (Ω)$ , including the interval branch (The subcritical W0 one p embedding into L2).

[F10]

For bounded open $Ω$ and $1 < p < \infty$ , $W_{0}^{1, p} (Ω)$ is a separable reflexive real Banach space (The zero-boundary W1p space is separable and reflexive).

[F11]

Under the strict PE12 exponent and Axiom of Choice, the flux-space definition uses $V = W_{0}^{1, p} (Ω)$ with gradient norm and $H = L^{2} (Ω)$ ; Poincaré [F7] makes this norm equivalent to the inherited Sobolev norm, so [F10] supplies separability and reflexivity. The compact embedding [F9] is dense, and the evolution-triple definition fixes their real pairing (Coercive monotone p-growth diffusion fluxes, Banach evolution triples and weak time derivatives).

[A1]

Countable Choice is the declared background for the finite-element Sobolev, density, and compactness interfaces used here (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

The PE12 evolution-triple application assumes the Axiom of Choice through the reflexivity, compact embedding, and flux-space interfaces (The Axiom of Choice).

Proof

1.1A1F1F8givenalgebra

The nodal system in [F1] has an invertible interior mass matrix whenever $V_{h} \neq {0}$ ; if $V_{h} = {0}$ , set $P_{h} f = 0$ . For finite $r$ , every basis function is bounded by a mesh-independent constant and is supported on the cells incident to its node. Hölder and the coefficient scaling in [F1] therefore give, for $b_{i} = h^{- d} \int_{Ω} f ϕ_{i}$ , $∣ b_{i} ∣^{r} \leq C h^{- d} \int_{supp ϕ_{i}} ∣ f ∣^{r}, ∥ b ∥_{ℓ^{r}} \leq C h^{- d / r} ∥ f ∥_{L^{r} (Ω)} .$ The second inequality follows because on each cell only the fixed number $N (d, k)$ of local basis supports occur, so their overlap is bounded. For $r = \infty$ , $∥ b ∥_{ℓ^{\infty}} \leq C ∥ f ∥_{\infty}$ .

1.2givenconstructalgebra

For $d = 1$ , the explicit meshes of $Ω = (0, 1)$ are $K_{j} = [j / N, (j + 1) / N]$ , $0 \leq j < N$ . They meet at whole endpoints, $h_{N} = 1 / N$ , each inradius is $1 / (2 N)$ , their shape ratio is $2$ , and $θ = 1$ ; hence $h_{N} \to 0$ . For $d \geq 2$ , in every grid cube $a + [0, 1 / N]^{d}$ , take the $d!$ simplices with cumulative vertices $a, a + N^{- 1} e_{π_{1}}, a + N^{- 1} (e_{π_{1}} + e_{π_{2}}), \dots, a + N^{- 1} \sum_{j = 1}^{d} e_{j}$ for permutations $π$ of ${1, \dots, d}$ . In coordinates $t = N (x - a)$ each is $1 \geq t_{π_{1}} \geq \dots \geq t_{π_{d}} \geq 0$ . Sorting coordinates covers the cube and strict orderings have disjoint interiors. For a permutation $π$ , the barycentric gaps are $g_{0} = 1 - t_{π_{1}}$ , $g_{j} = t_{π_{j}} - t_{π_{j + 1}}$ for $1 \leq j < d$ , and $g_{d} = t_{π_{d}}$ ; the simplex is $g_{j} \geq 0$ , $\sum_{j} g_{j} = 1$ . For two permutations, every pair of coordinates ordered oppositely must be equal in their intersection. In either order chain, that equality forces every intervening coordinate equal, hence sets the corresponding consecutive barycentric gaps to zero. The resulting equality blocks have the same relative order in both permutations: otherwise a pair of blocks would be oppositely ordered and would have been merged by an inversion equality. Thus the intersection is exactly the face obtained by setting those gaps to zero, so the within-cube triangulation is matching. On a cube face, coordinates fixed to $1$ occur first and those fixed to $0$ occur last in every compatible ordering; the induced ordering of the remaining coordinates is identical from either adjacent cube. Thus the global simplices meet only in whole faces and form a matching mesh.

2.1F1F8step 1.1algebra

If $V_{h} = {0}$ , then $P_{h} f = 0$ and the $L^{r}$ bounds are immediate. Otherwise, if $c$ is the coefficient vector of $P_{h} f$ , then $A_{h} c = b$ . Let $D = A_{h}^{- 1}$ , and let $R, C_{0}$ bound its absolute row and column sums from [F1]. For $1 < r < \infty$ , apply [F8] on the finite counting measure to the factors $a_{j}^{1 / r} ∣ b_{j} ∣$ and $a_{j}^{1 - 1 / r}$ , where $a_{j} = ∣ D_{i j} ∣$ . This gives $\sum_{j} a_{j} ∣ b_{j} ∣ \leq (\sum_{j} a_{j} ∣ b_{j} ∣^{r})^{1 / r} (\sum_{j} a_{j})^{1 - 1 / r}$ . If the row sum is zero both sides vanish. Raising to $r$ and summing the rows gives $\sum_{i} ∣ \sum_{j} D_{i j} b_{j} ∣^{r} \leq R^{r - 1} \sum_{i, j} ∣ D_{i j} ∣ ∣ b_{j} ∣^{r} \leq R^{r - 1} C_{0} \sum_{j} ∣ b_{j} ∣^{r} .$ For $r = 1$ , the triangle inequality and the column sums directly give $\sum_{i} ∣ \sum_{j} D_{i j} b_{j} ∣ \leq C_{0} \sum_{j} ∣ b_{j} ∣$ , the same displayed bound. For $r = \infty$ , the row-sum bound gives $∥ D b ∥_{ℓ^{\infty}} \leq R ∥ b ∥_{ℓ^{\infty}}$ . Combining these coefficient estimates with [F1] proves the asserted $L^{r}$ bound for $P_{h}$ . If $f \in L^{2}$ , the mass equations are exactly orthogonality to $V_{h}$ , so $P_{h}$ is the orthogonal projection and fixes every $w_{h} \in V_{h}$ .

2.2givenstep 1.2algebra

Each cube simplex contains opposite cube corners and lies in its cube, so its diameter is $h_{N} = \sqrt{d} / N$ . In row order $π_{1}, \dots, π_{d}$ , the cumulative-vertex edge matrix has columns $e_{π_{1}}, e_{π_{1}} + e_{π_{2}}, \dots, \sum_{j = 1}^{d} e_{π_{j}}$ before scaling by $1 / N$ . Subtracting each preceding column from the next gives the permuted identity matrix, so its determinant has absolute value $N^{- d}$ ; the simplex volume is $1 / (d! N^{d})$ . Its facets in ordered coordinates are $t_{π_{1}} = 1$ , $t_{π_{d}} = 0$ , and $t_{π_{j}} = t_{π_{j + 1}}$ . Put $r = 1 / (2 + (d - 1) \sqrt{2})$ and set $t_{π_{j}} = r + (d - j) \sqrt{2} r$ . The endpoint gaps are $r$ and successive coordinate gaps are $\sqrt{2} r$ , so this point is at perpendicular distance $r$ from every facet and is the center of an inscribed ball of radius $r$ . For any inscribed ball of radius $s$ , its center must have endpoint gaps at least $s$ and successive coordinate gaps at least $\sqrt{2} s$ ; adding these gaps gives $1 \geq 2 s + (d - 1) \sqrt{2} s$ . Thus the inradius is exactly $r / N$ and the shape ratio is $\sqrt{d} (2 + (d - 1) \sqrt{2})$ , independent of $N$ , all cell diameters equal $h_{N}$ (global quasi-uniformity with $θ = 1$ ), and only the $d!$ ordered shapes occur. The cube is a bounded Lipschitz polyhedral open set. At a boundary point choose a vector $v$ with $n_{j} \cdot v > 0$ for each active inward face normal $n_{j}$ , and write nearby points as $z + y + s v$ with $y ⊥ v$ . The active inequalities become $s > - n_{j} \cdot y / (n_{j} \cdot v)$ ; inactive face inequalities remain strict in a sufficiently small neighborhood. Thus locally the cube is the epigraph of the maximum of finitely many affine functions, which is Lipschitz. These are nonempty mesh families satisfying the $d \geq 2$ hypotheses.

3.1F2F3F7F8step 2.1algebra

Fix $1 < p < \infty$ and $v \in W_{0}^{1, p} (Ω)$ . For sufficiently fine meshes, since $P_{h} Q_{h} v = Q_{h} v$ , the inverse inequality in [F3] and the $L^{p}$ bound from step 2.1 give $∥ \nabla P_{h} v ∥_{p} \leq ∥ \nabla Q_{h} v ∥_{p} + C h^{- 1} ∥ P_{h} (v - Q_{h} v) ∥_{p} \leq ∥ \nabla Q_{h} v ∥_{p} + C h^{- 1} ∥ v - Q_{h} v ∥_{p} \leq C_{p} ∥ \nabla v ∥_{p},$ by [F2]. The inverse estimate holds on the interval branch as recorded in [F3]; on simplex meshes summing its cell estimates uses $h_{K} \geq θ h$ . A sequence with $h \to 0$ has only finitely many coarser initial meshes. On each such mesh, $P_{h}$ is finite rank and its coefficient functionals $v \mapsto \int_{Ω} v ϕ_{i}$ are bounded on $L^{p}$ ; hence its $W^{1, p}$ output norm is bounded by a mesh-dependent multiple of $∥ v ∥_{L^{p}}$ by [F8]. Poincaré [F7] bounds this by a multiple of $∥ \nabla v ∥_{p}$ . Enlarging $C_{p}$ over the finitely many initial meshes gives the stated uniform estimate.

4.1F2step 2.1step 3.1algebra

Step 2.1 and the approximation estimate of [F2] show $∥ P_{h} v - v ∥_{p} \leq C h ∥ \nabla v ∥_{p}$ . For gradients, write $P_{h} v - v = P_{h} (v - Q_{h} v) + (Q_{h} v - v)$ . Apply the uniform gradient bound proved in the preceding step to $v - Q_{h} v \in W_{0}^{1, p}$ and the strong approximation in [F2]; both gradient terms tend to zero. Hence $P_{h} v \to v$ in the full $W^{1, p}$ norm.

5.1A1F5step 2.1step 4.1algebra

Let $ψ \in C_{c}^{\infty} (Ω)$ . Since $Q_{h} ψ \in V_{h}$ and $P_{h} Q_{h} ψ = Q_{h} ψ$ , the $L^{2}$ bound in step 2.1 gives $∥ P_{h} ψ - ψ ∥_{2} \leq ∥ P_{h} (ψ - Q_{h} ψ) ∥_{2} + ∥ Q_{h} ψ - ψ ∥_{2} \leq (C_{2} + 1) ∥ Q_{h} ψ - ψ ∥_{W^{1, 2}} ⟶ 0$ by [F2]. For arbitrary $f \in L^{2} (Ω)$ , choose $ψ \in C_{c}^{\infty} (Ω)$ with $∥ f - ψ ∥_{2} < ε$ using [F5]. The $L^{2}$ orthogonal projection is a contraction, so ${lim sup}_{h \to 0} ∥ P_{h} f - f ∥_{2} \leq 2 ε .$ Letting $ε ↓ 0$ proves strong $L^{2}$ convergence for every $f \in L^{2} (Ω)$ .

6.1F2F4F6step 3.1step 4.1step 5.1step 1.2step 2.2given

On each explicit mesh family, $V_{h} \subset V$ : [F4] proves this for matching simplices and the interval membership argument is part of [F2]. The preceding estimates give the $H$ -orthogonal projectors, uniform $V$ -stability, and strong $V$ approximation; pair the spatial index $N$ with any time partitions whose maximum steps tend to zero. The resulting family satisfies all clauses of [F6], with no coupling between spatial and time refinement.

7.1A2F7F9F10F11step 6.1given∎

In the PE12 branch, [A2], [F7], [F9], [F10], and [F11] give the evolution triple with dense continuous compact inclusion. The explicit stable families from step 6.1 therefore instantiate the stated PE12 application for every strict $p$ in the Statement.

Remarks

The drawing shows only the $d = 2$ , $N = 1$ coordinate-order subdivision. The blue triangle is $K_{(1, 2)} = {0 \leq x_{2} \leq x_{1} \leq 1}$ , the orange triangle is $K_{(2, 1)} = {0 \leq x_{1} \leq x_{2} \leq 1}$ , and their red diagonal is the shared whole edge $x_{1} = x_{2}$ . The coordinate proof above establishes matching for all $d$ and $N$ ; no proof uses visual features of this illustration.

\begin{tikzpicture}[scale=3]
  \fill[blue!15] (0,0)--(1,0)--(1,1)--cycle;
  \fill[orange!20] (0,0)--(0,1)--(1,1)--cycle;
  \draw[thick] (0,0) rectangle (1,1);
  \draw[very thick,red] (0,0)--(1,1);
  \draw[->] (0,0)--(1.18,0) node[right] {$x_1$};
  \draw[->] (0,0)--(0,1.18) node[above] {$x_2$};
  \node at (0.72,0.28) {$K_{(1,2)}$};
  \node at (0.28,0.72) {$K_{(2,1)}$};
  \node[below left] at (0,0) {$(0,0)$};
  \node[below] at (1,0) {$(1,0)$};
  \node[left] at (0,1) {$(0,1)$};
  \node[above right] at (1,1) {$(1,1)$};
\end{tikzpicture}

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Nonlinear backward Euler Galerkin existence and energy bounds

Statement

Assume the Axiom of Choice.

Assume the hypotheses in Coercive monotone p-growth diffusion fluxes and an admissible stable conforming FEM sequence $(V_{k}, Π_{k}, P_{k})$ from Conforming FEM families with stable L2 projectors. Let $f \in L^{p^{'}} (0, T; V^{*})$ , $u_{0} \in H$ , and on each partition $P_{k} = (I_{k}^{n})_{n = 1}^{N_{k}}$ put $f_{k}^{n} = ∣ I_{k}^{n} ∣^{- 1} \int_{I_{k}^{n}} f (t) d t$ and $U_{k}^{0} = Π_{k} u_{0}$ . Then for each $k$ there exist $U_{k}^{n} \in V_{k}$ , $1 \leq n \leq N_{k}$ , satisfying ${(\frac{U_{k}^{n} - U_{k}^{n - 1}}{τ_{k}^{n}}, v_{k})}_{H} + ⟨ A (U_{k}^{n}), v_{k} ⟩ = ⟨ f_{k}^{n}, v_{k} ⟩ (v_{k} \in V_{k}) .$ Every such solution obeys a bound, independent of $k$ and the time steps, on $\max_{n} ∥ U_{k}^{n} ∥_{H}^{2} + \sum_{n} ∥ U_{k}^{n} - U_{k}^{n - 1} ∥_{H}^{2} + \sum_{n} τ_{k}^{n} ∥ U_{k}^{n} ∥_{V}^{p} + ∥ D_{k} ∥_{L^{p^{'}} (0, T; V^{*})}^{p^{'}},$ where $D_{k} = (U_{k}^{n} - U_{k}^{n - 1}) / τ_{k}^{n}$ on $I_{k}^{n}$ . The bound depends only on the flux constants, $p$ , $T$ , the family stability constant $C_{Π}$ , $∥ u_{0} ∥_{H}$ , $∥ f ∥_{L^{p^{'}} (V^{*})}$ , $∥ g ∥_{L^{p^{'}} (Ω)}$ , and $∥ h ∥_{L^{1} (Ω)}$ .

In particular, the same existence and energy conclusion holds on the fixed-degree uniform interval meshes of $(0, 1)$ in $d = 1$ and uniform Kuhn/Freudenthal simplex meshes of $(0, 1)^{d}$ for $d \geq 2$ , paired with any time partitions whose maximum step tends to zero independently of the spatial mesh.

Facts & Assumptions

Given: The Axiom of Choice, the stated flux and FEM hypotheses, $f \in L^{p^{'}} (0, T; V^{*})$ , $u_{0} \in H$ , and the time partitions.

[F1]

The flux is Carathéodory with the stated growth and coercivity conditions. Its induced map satisfies $∥ A (v) ∥_{V^{*}} \leq C_{1} (∥ g ∥_{L^{p^{'}}} + ∥ v ∥_{V}^{p - 1})$ , and obeys $⟨ A (v), v ⟩ \geq α ∥ v ∥_{V}^{p} - ∥ h ∥_{L^{1}}$ (Coercive monotone p-growth diffusion fluxes).

[F2]

The projections are $H$ -orthogonal, map $V$ into $V$ , and satisfy $\sup_{k} ∥ Π_{k} ∥_{V \to V} \leq C_{Π}$ (Conforming FEM families with stable L2 projectors).

[F3]

A continuous finite-dimensional map whose pairing with $x$ is positive on a sphere has a zero inside that sphere (An outward-pointing finite-dimensional map has a zero).

[F4]

In a real Hilbert space, $2 (x - y, x) = ∥ x ∥^{2} - ∥ y ∥^{2} + ∥ x - y ∥^{2}$ (The Hilbert-space discrete energy identity).

[F5]

Hölder bounds the cell averages of $f$ , and Young's inequality for conjugate $p, p^{'}$ gives, for $a, b \geq 0$ , $a b \leq \frac{α}{2} a^{p} + \frac{1}{p^{'}} {(\frac{2}{α})}^{1 / (p - 1)} b^{p^{'}} .$ (Holder's inequality for integrals, including the endpoint cases, Young's inequality for conjugate real exponents).

[F6]

The explicit interval and cube projector families satisfy the abstract uniform $V$ -stability and strong approximation clauses, and can be paired with arbitrary independent time partitions tending to zero (L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes).

[F7]

Dominated convergence passes pointwise limits through an integral under a fixed integrable majorant (Dominated convergence).

[A1]

The hypotheses are stated on the Axiom-of-Choice branch inherited from the flux definition; for fixed $k$ the grid is finite, so the proof selects only finitely many roots and does not use Countable Choice (The Axiom of Choice).

Proof

1.1A1F1F5F7givenconstructalgebra

Fix $k$ . If $V_{k} = {0}$ , then $U_{k}^{0} = Π_{k} u_{0} = 0$ , and setting $U_{k}^{n} = 0$ for every $n$ satisfies every equation (the only test vector is zero). Assume $V_{k} \neq {0}$ . For a time cell $I_{k}^{n}$ , the Bochner average $f_{k}^{n} \in V^{*}$ exists. Define $F_{n} : V_{k} \to V_{k}^{*}$ by $⟨ F_{n} (z), w_{k} ⟩ = τ_{k}^{- 1} (z - U_{k}^{n - 1}, w_{k})_{H} + ⟨ A (z) - f_{k}^{n}, w_{k} ⟩ .$ Choose a fixed $H$ -orthonormal basis $ψ_{1}, \dots, ψ_{N}$ of $V_{k}$ and identify $V_{k}^{*}$ with $R^{N}$ by its values on that basis. To prove continuity, let $z_{m} = \sum_{j} c_{m j} ψ_{j}$ with $c_{m j} \to c_{j}$ , and put $z = \sum_{j} c_{j} ψ_{j}$ . The coefficients are bounded by some $M$ , so $∣ \nabla z_{m} ∣ \leq M \sum_{j} ∣ \nabla ψ_{j} ∣ = : B \in L^{p} (Ω)$ and the gradients converge pointwise outside one null set. For each test basis vector $ψ_{i}$ , Carathéodory continuity gives pointwise convergence of $a (x, \nabla z_{m}) \cdot \nabla ψ_{i}$ , while growth bounds its absolute value by $C (g + B^{p - 1}) ∣ \nabla ψ_{i} ∣ \in L^{1}$ by Hölder [F5]. Thus [F7] proves convergence of every flux test coordinate. The mass and load coordinates are continuous and affine, so $F_{n}$ is a continuous finite-dimensional map, without a global operator-continuity assumption. Finite dimensionality gives constants $c_{k}, C_{k} > 0$ with $c_{k} ∥ z ∥_{H} \leq ∥ z ∥_{V} \leq C_{k} ∥ z ∥_{H}$ on $V_{k}$ . Coercivity [F1] therefore gives, with $r = ∥ z ∥_{H}$ , $⟨ F_{n} (z), z ⟩ \geq τ_{k}^{- 1} r^{2} + α c_{k}^{p} r^{p} - (τ_{k}^{- 1} ∥ U_{k}^{n - 1} ∥_{H} + C_{k} ∥ f_{k}^{n} ∥_{V^{*}}) r - ∥ h ∥_{L^{1}} .$ Since $p > 1$ , the right side is positive for all sufficiently large $r$ .

2.1A1F3step 1.1givenchoose

Apply [F3] on such a sphere in the chosen coordinates to obtain $F_{n} (U_{k}^{n}) = 0$ . Starting with $U_{k}^{0} = Π_{k} u_{0}$ , repeat this finite step for $n = 1, \dots, N_{k}$ ; this constructs the discrete solution on the partition.

3.1F1F2F4F5step 2.1givenalgebra

Test the equation with $v_{k} = U_{k}^{n}$ . By [F4], $\frac{1}{2} (∥ U_{k}^{n} ∥_{H}^{2} - ∥ U_{k}^{n - 1} ∥_{H}^{2} + ∥ U_{k}^{n} - U_{k}^{n - 1} ∥_{H}^{2}) + τ_{k}^{n} α ∥ U_{k}^{n} ∥_{V}^{p} \leq τ_{k}^{n} ∥ f_{k}^{n} ∥_{V^{*}} ∥ U_{k}^{n} ∥_{V} + τ_{k}^{n} ∥ h ∥_{L^{1}} .$ Use [F5] to absorb half the coercive term and sum over $n$ . Hölder on each cell gives $\sum_{n} τ_{k}^{n} ∥ f_{k}^{n} ∥_{V^{*}}^{p^{'}} \leq ∥ f ∥_{L^{p^{'}} (V^{*})}^{p^{'}}$ . Consequently $\frac{1}{2} ∥ U_{k}^{m} ∥_{H}^{2} + \frac{1}{2} \sum_{n \leq m} ∥ U_{k}^{n} - U_{k}^{n - 1} ∥_{H}^{2} + \frac{α}{2} \sum_{n \leq m} τ_{k}^{n} ∥ U_{k}^{n} ∥_{V}^{p} \leq \frac{1}{2} ∥ U_{k}^{0} ∥_{H}^{2} + T ∥ h ∥_{L^{1}} + C_{α, p} ∥ f ∥_{L^{p^{'}} (V^{*})}^{p^{'}},$ for every $m$ . Orthogonality gives $∥ U_{k}^{0} ∥_{H} \leq ∥ u_{0} ∥_{H}$ ; taking the maximum over $m$ bounds the first three quantities uniformly in $k$ .

4.1F1F2step 3.1givenalgebra

For $v \in V$ , orthogonality and $D_{k} ∣_{I_{k}^{n}} \in V_{k}$ give $⟨ D_{k}, v ⟩ = (D_{k}, i v)_{H} = (D_{k}, Π_{k} v)_{H} = ⟨ f_{k}^{n} - A (U_{k}^{n}), Π_{k} v ⟩ .$ Thus [F2] yields $∥ D_{k} (t) ∥_{V^{*}} \leq C_{Π} (∥ f_{k}^{n} ∥_{V^{*}} + ∥ A (U_{k}^{n}) ∥_{V^{*}})$ . The flux growth bound [F1] and the inequality $(a + b)^{p^{'}} \leq 2^{p^{'} - 1} (a^{p^{'}} + b^{p^{'}})$ give $∥ D_{k} ∥_{L^{p^{'}} (V^{*})}^{p^{'}} \leq C_{p, C_{1}} C_{Π}^{p^{'}} (∥ f ∥_{L^{p^{'}} (V^{*})}^{p^{'}} + T ∥ g ∥_{L^{p^{'}} (Ω)}^{p^{'}} + \sum_{n} τ_{k}^{n} ∥ U_{k}^{n} ∥_{V}^{p}) .$ Here $(p - 1) p^{'} = p$ , and $g$ is constant in time. Step 3.1 bounds the last sum, proving the derivative estimate with precisely the stated family and data dependencies.

5.1F6step 2.1step 3.1step 4.1given∎

For the fixed-degree interval and cube sequences in the Statement, [F6] verifies the abstract FEM projector hypotheses, including the independent time partitions. The preceding construction and energy estimates therefore apply to these concrete nonempty meshes.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Conditional monotone finite-element convergence

Statement

Assume the Axiom of Choice. Let $d \geq 1$ , let $Ω \subset R^{d}$ be a bounded open set, let $1 \max (1, 2 d / (d + 2))$ , and let $T > 0$ . Let $V = W_{0}^{1, p} (Ω; R)$ with gradient norm, $H = L^{2} (Ω; R)$ , and let $a (x, ξ)$ satisfy Coercive monotone p-growth diffusion fluxes. Assume an admissible stable conforming FEM sequence $(V_{k}, Π_{k}, P_{k})$ from Conforming FEM families with stable L2 projectors. For $f \in L^{p^{'}} (0, T; V^{*})$ and $u_{0} \in H$ , set $f_{k}^{n} = ∣ I_{k}^{n} ∣^{- 1} \int_{I_{k}^{n}} f$ and $U_{k}^{0} = Π_{k} u_{0}$ . For each $k$ , let $(U_{k}^{n})$ be any solution of the backward Euler equations in Nonlinear backward Euler Galerkin existence and energy bounds.

There are a subsequence, still indexed by $k$ , and a unique $u \in L^{p} (0, T; V) \cap L^{\infty} (0, T; H) \cap C ([0, T]; H)$ such that ${\overline{U}}_{k} \to u strongly in L^{r} (0, T; H) for every 1 \leq r < \infty, {\overline{U}}_{k} ⇀ u in L^{p} (0, T; V),$ the fluxes $a (x, \nabla {\overline{U}}_{k})$ converge weakly in $L^{p^{'}} ((0, T) \times Ω; R^{d})$ to $a (x, \nabla u)$ , and $u^{'} + A (u) = f in L^{p^{'}} (0, T; V^{*}), u (0) = u_{0} .$ Existence of each discrete solution and uniform energy bounds are included in Nonlinear backward Euler Galerkin existence and energy bounds. The abstract convergence claim is conditional on the FEM projector clauses. For $Ω = (0, 1)$ in $d = 1$ and $Ω = (0, 1)^{d}$ in each $d \geq 2$ , the explicit fixed-degree interval and Kuhn/Freudenthal simplex families of L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes verify those clauses for any independent time partitions with vanishing maximum step. These are concrete nonempty instances of the theorem on the full approved strict $p$ range.

Facts & Assumptions

Given: The Axiom of Choice; the stated domain, exponents, flux, projectors, partitions, load and initial datum; and a sequence of discrete solutions.

[F1]

The strict threshold gives the dense compact inclusion $W_{0}^{1, p} (Ω) ⋐ L^{2} (Ω)$ , and the flux definition fixes $V, H, V^{*}$ and the canonical pairings (The subcritical W0 one p embedding into L2, Coercive monotone p-growth diffusion fluxes, Banach evolution triples and weak time derivatives).

[F14]

The Carathéodory composition lemma makes the flux evaluated at a measurable vector field measurable (A Caratheodory integrand composed with measurable functions is measurable).

[F2]

The FEM sequence has $H$ -orthogonal projections bounded uniformly $V \to V$ , converging strongly to the identity on $V$ , with maximum time step tending to zero (Conforming FEM families with stable L2 projectors).

[F3]

The nonlinear backward Euler equations have solutions and those solutions have uniform $L^{p} (V)$ , $L^{\infty} (H)$ , and $L^{p^{'}} (V^{*})$ discrete derivative bounds (Nonlinear backward Euler Galerkin existence and energy bounds).

[F4]

Bochner interval averages on partitions with vanishing maximum cell length converge strongly to the load in $L^{p^{'}} (0, T; V^{*})$ (Time translations and mollification in Bochner Lp).

[F5]

The piecewise-constant reconstructions with these bounds are relatively compact in $L^{p} (0, T; H)$ ; a uniform $L^{\infty} (H)$ bound upgrades the same subsequence to every finite $L^{r} (H)$ (Piecewise-constant discrete Aubin–Lions compactness).

[F6]

A sequence bounded in $L^{p} (0, T; W_{0}^{1, p})$ and convergent strongly in $L^{p} (0, T; H)$ has a subsequence weakly convergent in the Bochner Sobolev space, with its space-time gradients weakly convergent (Weak compactness of bounded space-time Sobolev gradients).

[F7]

Scalar real $L^{p^{'}}$ is reflexive, and under the ultrafilter lemma, Dependent Choice and Hahn–Banach, bounded sequences in reflexive Banach spaces have weakly convergent subsequences (Reflexivity of Lp for one less p less infinity, Reflexivity is equivalent to weak subsequential compactness of bounded sequences, Reflexivity is surjectivity of the canonical map).

[F8]

Functions in $L^{p} (V)$ whose derivative is in $L^{p^{'}} (V^{*})$ and which are bounded in $L^{\infty} (H)$ have a continuous $H$ representative and obey the integrated energy identity (The Banach evolution-space energy identity and H trace).

[F9]

If the weak gradient limit and weak flux limit satisfy the Minty limsup inequality, monotonicity and the growth bound identify the flux as $a (x, \nabla u)$ almost everywhere (Minty identification for a monotone Carathéodory flux, Coercive monotone p-growth diffusion fluxes).

[F10]

The Hilbert discrete energy identity is $2 (x - y, x) = ∥ x ∥^{2} - ∥ y ∥^{2} + ∥ x - y ∥^{2}$ (The Hilbert-space discrete energy identity, Hilbert space).

[F11]

Hölder makes the variational pairings of $L^{p^{'}} (V^{*})$ and $L^{p} (V)$ integrable (Holder's inequality for integrals, including the endpoint cases).

[F12]

A scalar $W^{1, 1} (0, T)$ class has an absolutely continuous representative on $[0, T]$ , so its integration-by-parts formula includes the endpoint values (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives).

[F13]

The fixed-degree zero-boundary Lagrange spaces on the uniform Kuhn meshes of $(0, 1)^{d}$ , $d \geq 2$ , and on uniform interval partitions of $(0, 1)$ , $d = 1$ , have $L^{2}$ projectors uniformly bounded on $W_{0}^{1, p}$ and converging strongly there for every $1 < p < \infty$ ; the same projections converge strongly on $L^{2}$ (L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes).

[A1]

Axiom of Choice supplies Countable and Dependent Choice, the ultrafilter lemma and Hahn–Banach through the cited choice interfaces; these are used in compact Sobolev embedding and weak subsequence extraction (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, Hahn-Banach dominated extension theorem for real vector spaces, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The ultrafilter extension principle (UL/BPI), The real dominated-extension principle as an additional hypothesis over ZF).

Proof

1.1F1F2F4givenalgebra

The orthogonal projections are contractions on $H$ , and for any $v \in V$ , $Π_{k} v \to v$ in $V$ and hence in $H$ . Given $u_{0} \in H$ , choose $v \in V$ close to $u_{0}$ using density in [F1]. Then $∥ Π_{k} u_{0} - u_{0} ∥_{H} \leq 2 ∥ u_{0} - v ∥_{H} + ∥ Π_{k} v - v ∥_{H} \to 2 ∥ u_{0} - v ∥_{H} .$ Letting $v$ approach $u_{0}$ proves $Π_{k} u_{0} \to u_{0}$ in $H$ . Also [F4] gives $f_{k} \to f$ strongly in $L^{p^{'}} (V^{*})$ for the piecewise-constant averages $f_{k} (t) = f_{k}^{n}$ on $I_{k}^{n}$ .

1.2A1F3F5F6given

By [F3], the reconstructions ${\overline{U}}_{k}$ are uniformly bounded in $L^{p} (V)$ and $L^{\infty} (H)$ , and their discrete derivatives $D_{k}$ are uniformly bounded in $L^{p^{'}} (V^{*})$ . The maximum partition step tends to zero, so [F5] gives a subsequence converging strongly in $L^{p} (H)$ and, by the $L^{\infty} (H)$ bound, in every finite $L^{r} (H)$ . Applying [F6] to this subsequence gives a further subsequence and $u \in L^{p} (V)$ such that ${\overline{U}}_{k} ⇀ u$ in $L^{p} (V)$ , its image in $L^{p} (H)$ is the strong limit, and $\nabla {\overline{U}}_{k} ⇀ \nabla u$ in $L^{p} (Q; R^{d})$ .

1.3F1F8givenalgebra

If $u$ and $\tilde{u}$ are two solutions with the same data, their difference $w = u - \tilde{u}$ has zero initial value and satisfies $w^{'} + A (u) - A (\tilde{u}) = 0$ . By [F8], for every $t$ , $\frac{1}{2} ∥ w (t) ∥_{H}^{2} + \int_{0}^{t} ⟨ A (u) - A (\tilde{u}), w ⟩ = 0.$ Pointwise monotonicity in [F1] makes the integral nonnegative, so $w (t) = 0$ for every $t$ . This proves uniqueness in the abstract conclusion.

2.1A1F1F14F7step 1.2algebra

Put $q_{k} (t, x) = a (x, \nabla {\overline{U}}_{k} (t, x))$ . The gradients are measurable, so [F14] and the growth hypothesis [F1] give $∥ q_{k} ∥_{L^{p^{'}} (Q)} \leq C (T^{1 / p^{'}} ∥ g ∥_{L^{p^{'}} (Ω)} + ∥ \nabla {\overline{U}}_{k} ∥_{L^{p} (Q)}^{p - 1}),$ so $(q_{k})$ is bounded in vector-valued $L^{p^{'}} (Q)$ . Each of its $d$ coordinates is bounded in scalar $L^{p^{'}} (Q)$ and hence has a weakly convergent subsequence by [F7]. Successive extraction over the finite coordinate set gives one subsequence weakly convergent in every coordinate; restriction of any bounded functional on the finite product to its coordinate embeddings shows weak convergence of the vector fields. Write the limit as $q \in L^{p^{'}} (Q; R^{d})$ .

3.1F2F4F5F11step 1.2step 2.1given

For each fixed $v \in V$ and $ϕ \in C_{c}^{\infty} (0, T)$ , integrate the discrete equation against $ϕ$ and test with $Π_{k} v$ . If ${\hat{U}}_{k}$ is the piecewise-affine reconstruction, integration by parts gives $- \int_{0}^{T} ({\hat{U}}_{k}, Π_{k} v)_{H} ϕ^{'} + \int_{Q} q_{k} \cdot \nabla Π_{k} v ϕ = \int_{0}^{T} ⟨ f_{k}, Π_{k} v ⟩ ϕ .$ On each cell, $∥ {\hat{U}}_{k} - {\overline{U}}_{k} ∥_{V^{*}} \leq τ_{k}^{n} ∥ D_{k} ∥_{V^{*}}$ , hence $∥ {\hat{U}}_{k} - {\overline{U}}_{k} ∥_{L^{p^{'}} (0, T; V^{*})} \leq δ_{k} ∥ D_{k} ∥_{L^{p^{'}} (0, T; V^{*})} \to 0.$ Now [F2] gives $Π_{k} v \to v$ in $V$ ; steps 1.2 and 2.1 give the strong state convergence and weak flux convergence; [F4] gives strong convergence of $f_{k}$ . Hölder [F11] therefore permits passage in every term, yielding $- \int_{0}^{T} (u, v)_{H} ϕ^{'} + \int_{Q} q \cdot \nabla v ϕ = \int_{0}^{T} ⟨ f, v ⟩ ϕ .$ Thus the distributional derivative of $u$ is $f - B_{q}$ , where $⟨ B_{q} (t), v ⟩ = \int_{Ω} q (t, x) \cdot \nabla v (x) d x$ and $B_{q} \in L^{p^{'}} (0, T; V^{*})$ .

4.1F3F8step 1.2step 3.1algebra

The uniform bound $∥ {\overline{U}}_{k} (t) ∥_{H} \leq M$ and strong $L^{p} (H)$ convergence imply $(∥ u (t) ∥_{H} - M)_{+} \leq ∥ u (t) - {\overline{U}}_{k} (t) ∥_{H}$ almost everywhere. Taking $L^{p}$ norms and passing to the limit shows $u \in L^{\infty} (0, T; H)$ . Step 3.1 places $u$ in $L^{p} (V)$ with derivative $f - B_{q} \in L^{p^{'}} (V^{*})$ ; [F8] therefore gives its continuous $H$ representative and energy identity.

5.1A1F2F11step 3.1step 4.1given

Fix $v \in V$ . The scalar affine interpolants $α_{k} (t) = ({\hat{U}}_{k} (t), v)_{H}$ are uniformly bounded, and $α_{k}^{'} (t) = ⟨ D_{k} (t), v ⟩$ is uniformly bounded in $L^{p^{'}}$ . Hölder gives a common modulus $∣ α_{k} (t) - α_{k} (s) ∣ \leq C_{v} ∣ t - s ∣^{1 / p}$ . The reconstruction defect in step 3.1 and the strong state convergence imply $α_{k} \to α : = (u, v)_{H}$ in $L^{1} (0, T)$ ; $α$ is continuous by step 4.1. This convergence is uniform: if $∣ α_{k} (t_{k}) - α (t_{k}) ∣ \geq ε$ along a subsequence, the common modulus and continuity of $α$ give a fixed $δ > 0$ on which the difference is at least $ε / 3$ around $t_{k}$ . The intersected interval has length at least $\min (δ, T)$ , contradicting $L^{1}$ convergence. Therefore $(U_{k}^{N_{k}}, v)_{H} = α_{k} (T) \to (u (T), v)_{H}$ .

5.2F1F2F4F11F12step 1.1step 3.1step 4.1given

Use a smooth $ϕ$ with $ϕ (0) = 1$ and $ϕ (T) = 0$ . Integrating the discrete equation by parts in time gives $- (U_{k}^{0}, Π_{k} v)_{H} - \int_{0}^{T} ({\hat{U}}_{k}, Π_{k} v)_{H} ϕ^{'} + \int_{Q} q_{k} \cdot \nabla Π_{k} v ϕ = \int_{0}^{T} ⟨ f_{k}, Π_{k} v ⟩ ϕ .$ The convergence arguments of step 3.1 and $Π_{k} u_{0} \to u_{0}$ from step 1.1 pass this identity to the limit. By step 3.1 the distributional derivative of $t \mapsto (u (t), v)_{H}$ is $⟨ f - B_{q}, v ⟩ \in L^{1} (0, T)$ ; [F12] gives its absolutely continuous representative, which agrees with the continuous $H$ representative from step 4.1. Its integration-by-parts formula gives the same limit identity with $u (0)$ in place of $u_{0}$ . Therefore $(u (0), v)_{H} = (u_{0}, v)_{H}$ for every $v \in V$ , and density of $V$ in [F1] proves $u (0) = u_{0}$ .

6.1F1F3step 4.1step 5.1algebra

The nodal values are uniformly bounded in $H$ by [F3]. Step 5.1 gives $U_{k}^{N_{k}} ⇀ u (T)$ against every $v \in V$ ; density of $V$ in $H$ and the uniform $H$ bounds extend this to weak convergence in $H$ . Consequently $∥ u (T) ∥_{H}^{2} \leq {lim inf}_{k} ∥ U_{k}^{N_{k}} ∥_{H}^{2}$ : if $u (T) \neq 0$ , test the weak convergence against $u (T)$ and apply Cauchy–Schwarz; the zero case is immediate.

7.1F4F8F10step 1.1step 1.2step 3.1step 4.1step 6.1algebra

Testing each discrete equation with $U_{k}^{n}$ , summing the exact identity [F10], and dropping the nonnegative jump sum gives $\int_{Q} q_{k} \cdot \nabla {\overline{U}}_{k} \leq \int_{0}^{T} ⟨ f_{k}, {\overline{U}}_{k} ⟩ + \frac{1}{2} ∥ U_{k}^{0} ∥_{H}^{2} - \frac{1}{2} ∥ U_{k}^{N_{k}} ∥_{H}^{2} .$ Strong convergence $f_{k} \to f$ and weak convergence ${\overline{U}}_{k} ⇀ u$ make the first term on the right converge to $\int_{0}^{T} ⟨ f, u ⟩$ . Steps 1.1 and 6.1 therefore imply ${lim sup}_{k} \int_{Q} q_{k} \cdot \nabla {\overline{U}}_{k} \leq \int_{0}^{T} ⟨ f, u ⟩ + \frac{1}{2} ∥ u_{0} ∥_{H}^{2} - \frac{1}{2} ∥ u (T) ∥_{H}^{2} .$ The continuous equation from step 3.1 and the energy identity in step 4.1 make the right-hand side equal to $\int_{Q} q \cdot \nabla u$ .

8.1F1F9step 1.2step 2.1step 3.1step 7.1

Apply [F9] to the weak gradient convergence in step 1.2, the weak flux convergence in step 2.1, and the limsup inequality in step 7.1. It gives $q = a (x, \nabla u)$ almost everywhere. Hence $B_{q} = A (u)$ and step 3.1 is the claimed weak evolution equation.

9.1F13givenstep 1.1step 1.3∎

For the unit interval in $d = 1$ , [F13] supplies the projector sequence for every $p > 1$ , which is the stated strict threshold. For the unit cube in any $d \geq 2$ , [F13] supplies it for every $p > \max (1, 2 d / (d + 2))$ . Choose any time partitions with maximum step tending to zero, independently of the spatial index. The resulting mesh/projector/time sequences satisfy every hypothesis used in the preceding convergence proof, so the abstract convergence and uniqueness conclusion holds on these explicit nonempty standard-mesh families without an additional projector assumption.

5 · Examples, counterexamples and false statements

None yet.