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Uniform annular Schauder bootstrap for spacelike graph equations

Statement

Work in ZFC. Let $n \geq 2$ be a fixed integer, $A = {y \in R^{n} : 1 / 2 < ∣ y ∣ < 4}$ and $A^{'} = {y \in R^{n} : 1 < ∣ y ∣ < 2}$ , and let $u_{R}$ be a family of smooth scalar graph heights on $R A$ , for $R \geq R_{0} \geq 1$ . Indices $i, j$ range from $1$ to $n$ , and repeated indices are summed. Set $v_{R} (y) = u_{R} (R y)$ and $w_{R} = v_{R} / R$ . Fix $0 < α < 1$ . The norms below are ordinary fixed-domain Hölder norms.

The general realized-coefficient version is as follows: if $a_{R}^{i j} (y) (v_{R})_{i j} + b_{R}^{j} (y) (v_{R})_{j} + c_{R} (y) v_{R} = f_{R} (y)$ , the symmetric matrices $a_{R}$ have common positive ellipticity bounds, $a_{R}, b_{R}, c_{R}$ have common $C^{2, α}$ bounds on $A$ , and $∥ f_{R} ∥_{C^{2, α} (A)} \leq K / R$ , then $\sum_{j = 0}^{4} R^{j} \sup_{R A^{'}} ∣ D_{x}^{j} u_{R} ∣ \leq C (\sup_{R A} ∣ u_{R} ∣ + R^{- 1}) .$

For a stationary spacelike maximal graph, with flux $P_{R}$ taking values in $R^{n}$ , the following explicit operator hypotheses supply those realized coefficient bounds, rather than assuming them from smoothness: ${div}_{y} P_{R} (y, D w_{R}) = 0$ , with no dependence on height; $∣ D w_{R} ∣ \leq p_{0}$ , $∥ w_{R} ∥_{\infty} \leq M_{0}$ ; on the whole slope ball $∣ p ∣ \leq p_{0}$ , $P_{R, p}$ is symmetric with $λ I \leq P_{R, p} \leq Λ I$ and $P_{R}$ has common joint $C^{4, α}$ bounds on slightly larger slope neighborhoods. Write $s_{R} (y, p) = \sum_{i} \partial_{y_{i}} P_{R}^{i} (y, p)$ , with partial derivatives at fixed $p$ , and assume $∥ s_{R} (\cdot, 0) ∥_{C^{2, α} (A)} \leq K / R^{2}$ . Then the same estimate follows. Uniformity over additional parameters means common constants on the specified parameter set; no claim of uniformity at a degenerate parameter endpoint is made. The stationary Lorentzian four-manifold application uses $n = 3$ ; its metric flux and sufficient background hypotheses are explicit in the flux lemma. Every application must verify these hypotheses, not only pointwise decay of the evaluated principal coefficients.

Facts & Assumptions

[F1]

The fixed-annulus rescaling and finite-cover identities hold. Annular scaling and fixed interior covers

[F2]

The explicit stationary Lorentzian flux has symmetric elliptic gradient derivative under its stated metric and spacelike-cone hypotheses. Ellipticity and zero-slope forcing of the stationary maximal flux

[F3]

For a smooth flux under the gradient lemma's ellipticity, slope and flux-position-derivative hypotheses, the differentiated flux equation gives an initial gradient Hölder estimate before any Schauder application. Gradient Hölder control by differentiated flux and one-dimensional lifting

[F4]

Under its inherited smooth-flux hypotheses, the preliminary bootstrap gives common realized coefficient bounds using only structural data and a height bound for the normalized graph. Uniform coefficient regularity before the final graph estimate

[F5]

The linear fourth-order estimate is linear in the height and the forcing once realized coefficient bounds are known. Linear fourth derivative bound with controlled realized coefficients

[F6]

Uniform asymptotic estimates require a common chart, radius and constants. Differentiated asymptotic bounds and parameter uniformity

[F7]

The graph type, maximality convention and admissible spacelike image require explicit hypotheses. Normal graphs and the nonlinear mean-curvature map

[F8]

Use ordinary Euclidean Hölder norms on the fixed annuli. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F9]

In dimension $n \geq 2$ and under Countable Choice, the interior Schauder estimate takes a bounded classical $C^{2}$ solution of a uniformly elliptic equation with controlled $C^{0, γ}$ coefficients and forcing to an interior $C^{2, γ}$ bound. Here $0 < γ < 1$ ; lower-order coefficients can be zero. Interior Schauder estimate for uniformly elliptic equations

[F10]

Work in ZFC, carrying the analytic suppliers' choice assumptions. Full Choice supplies Countable Choice: for a sequence of nonempty sets, choose on its family of values and compose the choice function with that sequence. The Axiom of Choice

[F11]

For totally differentiable maps between open Euclidean sets, $D (G \circ Φ) = D G (Φ) D Φ$ . The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

[F12]

Under Countable Choice, a continuous Banach-valued curve on $[a, b]$ whose interior derivative extends continuously to $[a, b]$ satisfies $φ (b) - φ (a) = \int_{a}^{b} φ^{'} (t) d t$ . Use the real scalar case $X = R$ and $[a, b] = [0, 1]$ . Fundamental theorem of calculus for Banach-valued continuous curves

[F13]

Under Countable Choice, $H^{1} = W^{1, 2}$ consists of square-integrable almost-everywhere classes with square-integrable first weak derivatives. For a bounded measurable uniformly elliptic real matrix $B$ , the homogeneous local weak equation means $z \in H^{1} (U)$ and $\int_{U} (B D z) \cdot D φ = 0$ for every $φ \in C_{c}^{\infty} (U)$ . Use the real scalar convention with zero lower-order coefficients and datum. The notation Hᵏ and the reserved zero-boundary symbol Local weak solutions of a divergence-form operator

[F14]

A continuous real integrand on a compact rectangle with a continuous parameter derivative may be differentiated through its Riemann integral. Apply twice, componentwise, with other spatial coordinates fixed. A uniform difference bound $η$ between integrable functions on $[0, 1]$ bounds the difference of their integrals by $η$ . Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error

[F15]

The real mean value theorem applies to continuous functions on a closed interval differentiable in its interior. Along line segments, a bound on a derivative therefore bounds increments by the length times that bound. The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)

[F16]

Under Countable Choice, for a bounded $C^{1}$ domain $U \subset R^{d}$ , $d \geq 2$ , and a vector field $V \in C^{1} (\overline{U}; R^{d})$ , the divergence theorem gives $\int_{U} div V = \int_{\partial U} V \cdot ν$ . In particular the integral is zero when $V$ vanishes near the boundary. Divergence on a bounded C1 Euclidean domain

[F20]

Use the homogeneous De Giorgi–Nash estimate recorded in External input, only after verifying the lifted weak equation in dimension $n + 1$ .

Proof

Given: The hypotheses in the Statement.

Goal: Establish the asserted estimate with the stated uniform dependence.

1.1givenF1F5F8F9F10

In the general realized-coefficient case apply [F5] to $v_{R}$ on $A$ with target annulus $A^{'}$ . It yields $∥ v_{R} ∥_{C^{4, α} (A^{'})} \leq C (∥ v_{R} ∥_{\infty} + K / R)$ . Rescale derivatives using [F1]. Thus the full original annular estimate follows whenever the stated coefficient hypothesis means actual $C^{2, α}$ bounds as functions of position.

1.2givenF1F3F4F8F10

If $P_{R}$ is smooth, [F3] and [F4] give the preliminary coefficient bounds under the stated structural hypotheses. For a general jointly $C^{4, α}$ flux, their smooth-flux interfaces are not invoked. Instead the following local argument extends the gradient and coefficient bootstrap to that finite regularity while keeping $w_{R}$ smooth, as already assumed. Choose a fixed finite sequence of nested annuli strictly between $A^{'}$ and $A$ , with room for all the successive interior estimates below. Work on balls with compact closures in the larger annuli, then cover the smaller ones using [F1]. All radii, separations and finite covering constants are fixed independently of $R$ . Suppress $R$ in the local calculation.

1.3F8F11F15algebra

On a convex ball, subtracting products gives $[f g]_{0, γ} \leq ∥ f ∥_{\infty} [g]_{0, γ} + ∥ g ∥_{\infty} [f]_{0, γ}$ for $0 < γ < 1$ ; iterate for finite products. The identity $Δ (f g) = f Δ g + g Δ f + Δ f Δ g$ , divided by the increment and passed to the limit, gives the product derivative rule. By [F11] and [F15] on line segments, an outer map with bounded first derivatives is Lipschitz on the ball times the slope ball. Hence $[G \circ Ψ]_{0, γ} \leq Lip (G) [Ψ]_{0, γ}$ . If $G$ is merely $γ$ -Hölder and $Ψ$ is Lipschitz, instead use $[G \circ Ψ]_{0, γ} \leq [G]_{0, γ} Lip (Ψ)^{γ}$ . These inequalities give the product and composition bounds used below.

1.4givenF11algebra

For each coordinate $k$ , put $z = w_{k}$ , $a (y) = P_{p} (y, D w (y))$ and $F (y) = - P_{y_{k}} (y, D w (y))$ , with the partial derivative at fixed slope. Differentiating the given flux equation gives $div (a D z) = div F$ . This differentiation uses only two joint derivatives of $P$ and the assumed smoothness of $w$ . In particular, $a$ is symmetric with $λ I \leq a \leq Λ I$ , $∣ z ∣ \leq p_{0}$ , and $∣ F ∣$ is bounded by the common structural bound. No uniform bound on $D^{2} w$ has been used.

2.1step 1.4F10F13F16F20algebra

On the chosen ball let $M = ∥ z ∥_{\infty} + ∥ F ∥_{\infty}$ . If $M = 0$ , the gradient component is zero and needs no estimate. Otherwise define $Z (y, t) = z (y) / M + t$ , $H (y) = F (y) / M$ , and on the ball times $(- 1, 1)$ set $B = (\begin{array}{cc} a & - H \\ - H^{T} & 1 + 2 / λ \end{array})$ . Since $∣ H ∣ \leq 1$ , its quadratic form is at least $(λ / 2) ∣ ξ ∣^{2} + ∣ τ ∣^{2}$ , using $2 ∣ τ H \cdot ξ ∣ \leq (λ / 2) ∣ ξ ∣^{2} + (2 / λ) τ^{2}$ ; an upper bound depends only on $λ, Λ$ . Independence of $t$ gives ${div}_{y, t} (B D Z) = {div}_{y} (a D z / M - H) = 0$ . The entries are $C^{1}$ locally, so this identity is classical. On every ball $U$ with compact closure in the cylinder, $Z$ and its first derivatives are bounded on $\overline{U}$ and are square-integrable because $U$ is bounded. To check the weak identities, let $φ \in C_{c}^{\infty} (U)$ and $1 \leq ℓ \leq n + 1$ . The ball is a bounded $C^{1}$ domain in dimension $n + 1 \geq 2$ . Both $Z φ e_{ℓ}$ and $φ B D Z$ are $C^{1}$ on a neighborhood of $\overline{U}$ and vanish near $\partial U$ , since the coefficients are $C^{1}$ there and $φ$ has compact support in $U$ . With Countable Choice supplied by [F10], [F16] and the product rule give $0 = \int_{U} div (Z φ e_{ℓ}) = \int_{U} (Z \partial_{ℓ} φ + φ \partial_{ℓ} Z)$ . These identities identify the classical derivatives as weak derivatives, so $Z \in H^{1} (U)$ by [F13]. Applying [F16] to the second field gives $0 = \int_{U} div (φ B D Z) = \int_{U} (B D Z) \cdot D φ + \int_{U} φ div (B D Z) = \int_{U} (B D Z) \cdot D φ$ . Thus [F13] verifies exactly the weak-solution hypotheses of [F20].

3.1step 2.1F1F8F10F20given

Apply [F20] after translation and dilation of the $(n + 1)$ -dimensional interior balls to unit balls. On the fixed cylinder $∣ Z ∣ \leq 2$ , so its $L^{2}$ norm on each such ball is uniformly bounded. Restrict the resulting estimate to $t = 0$ , multiply by the uniformly bounded $M$ , and cover a smaller $y$ ball by finitely many such patches. Performing this for all $k$ gives a uniform $C^{0, β}$ bound for $D w$ . Decrease the exponent if necessary to ensure $0 < β \leq α$ and $β < 1$ ; on fixed bounded sets this costs only a diameter factor. The exponent and constants depend only on the structural data and fixed separations.

4.1step 1.3step 3.1F8F9F10F11given

Put $S (y) = s (y, D w (y))$ . The chain rule gives $a^{i j} w_{i j} = - S$ . From step 3.1, the joint derivative bounds on $P$ and step 1.3, $a$ and $S$ have common $C^{0, β}$ bounds on interior balls. The matrix has the inherited ellipticity bounds, $w$ is smooth and bounded by $M_{0}$ , both lower-order coefficients are zero, and [F10] supplies Countable Choice. Applying [F9] with exponent $β$ gives a uniform $C^{2, β}$ bound for $w$ on smaller balls.

5.1step 1.3step 4.1F8F9F10F11givenalgebra

The chain and product rules with step 4.1 now give uniform $C^{1, β}$ bounds for $a$ and $S$ . Differentiating the equation once yields $a^{i j} (w_{k})_{i j} = - D_{k} S - (D_{k} a^{i j}) w_{i j}$ . Its right-hand side has a uniform $C^{0, β}$ bound by step 1.3, while $w_{k}$ is smooth and has a bounded supremum. Apply [F9] to each $w_{k}$ on smaller balls. This gives a uniform $C^{3, β}$ bound for $w$ , without an assumed uniform bound on its third derivatives.

6.1step 1.3step 5.1F8F9F10F11givenalgebra

Now $a$ and $S$ have uniform $C^{2, β}$ bounds, using joint derivatives of $P$ through order three and step 5.1. Twice differentiating the equation gives $a^{i j} (w_{k ℓ})_{i j} = - D_{k ℓ} S - (D_{k ℓ} a^{i j}) w_{i j} - (D_{k} a^{i j}) w_{i j ℓ} - (D_{ℓ} a^{i j}) w_{i j k}$ . Every term on the right has a uniform $C^{0, β}$ bound; $w_{k ℓ}$ is smooth with bounded supremum. Apply [F9] once more to obtain a uniform $C^{4, β}$ bound for $w$ on further smaller balls. The differentiated equations use only the existing finite derivatives of $P$ ; the classical $C^{2}$ membership needed in these applications comes from the given smoothness of $w$ . All preliminary bounds may depend on $M_{0}$ , without being linear in it.

7.1step 1.3step 6.1F8F11F15given

In particular all derivative suprema of $w$ through order four are uniformly bounded. By [F15], $D w, D^{2} w, D^{3} w$ are uniformly Lipschitz on convex interior balls. For $0 \leq t \leq 1$ set $Φ_{t} (y) = (y, t D w (y))$ , $G (y, p) = P_{p} (y, p)$ and $H_{j} (y, p) = s_{p_{j}} (y, p)$ . The maps $Φ_{t}, D Φ_{t}, D^{2} Φ_{t}$ are uniformly bounded and Lipschitz. For either $J = G$ or $J = H_{j}$ , [F11] twice gives $D^{2} (J \circ Φ_{t}) = D^{2} J (Φ_{t}) [D Φ_{t}, D Φ_{t}] + D J (Φ_{t}) D^{2} Φ_{t}$ . For $G$ , joint derivatives through order two use derivatives of $P$ through order three; for $H_{j}$ they use derivatives of $P$ through order four. Their $α$ -Hölder bounds follow from the joint $C^{4, α}$ hypothesis, using [F15] for lower derivatives. Thus step 1.3 gives a common $C^{2, α}$ bound for $a = G \circ Φ_{1}$ and for $h_{j} (y, t) = H_{j} (Φ_{t} (y))$ uniformly in $t$ . No fifth derivative of $P$ is used.

8.1step 7.1F1F8F14given

On closed coordinate boxes inside these balls, $h_{j}$ and its first two spatial derivatives are continuous jointly with $t \in [0, 1]$ , including at the boundary of the slope ball because $P$ is defined on a slightly larger slope neighborhood. By [F14], twice passing spatial derivatives through the integral gives $D_{y}^{ν} b^{j} (y) = \int_{0}^{1} D_{y}^{ν} h_{j} (y, t) d t$ for $∣ ν ∣ \leq 2$ , where $b^{j} = \int_{0}^{1} h_{j} d t$ . The uniform integral error bound in [F14] bounds the derivative suprema and the second-derivative $α$ -Hölder differences by the bounds of step 7.1. Finite interior covers assemble common $C^{2, α}$ bounds for $a$ and $b$ on a fixed intermediate annulus: close pairs lie in a controlled ball, and separated pairs use twice the supremum divided by the fixed separation to the power $α$ , as in [F1]. This completes the finite-regularity coefficient bootstrap.

9.1step 1.1step 8.1F1F5F10F11F12givenalgebra

Apply [F11] to the composition $y \mapsto P_{R} (y, D w_{R} (y))$ to expand the flux equation as $A_{R} : D^{2} w_{R} + s_{R} (y, D w_{R}) = 0$ . For each fixed $y$ , the curve $φ_{y} (t) = s_{R} (y, t D w_{R} (y))$ is continuous on $[0, 1]$ , and [F11] gives $φ_{y}^{'} (t) = \sum_{j} (s_{R})_{p_{j}} (y, t D w_{R} (y)) (w_{R})_{j} (y)$ . This derivative extends continuously to both endpoints: the whole segment lies in the slope ball, and the assumed joint regularity holds on a neighborhood of it. Thus [F12], with Countable Choice supplied by [F10], gives $s_{R} (y, D w_{R}) = s_{R} (y, 0) + b_{R} \cdot D w_{R}$ . Multiplying by $R$ gives $A_{R} : D^{2} v_{R} + b_{R} \cdot D v_{R} = - R s_{R} (y, 0)$ . The forcing has $C^{2, α}$ norm at most $K / R$ . On the intermediate annulus of step 8.1 the realized coefficients have common $C^{2, α}$ bounds, so apply [F5] there with target $A^{'}$ exactly as in step 1.1, and use [F1] to rescale back. The supremum on that intermediate annulus is bounded by $\sup_{A} ∣ v_{R} ∣$ . This proves the asserted linear height estimate.

10.1F2F6F7step 9.1∎

For stationary metric applications, [F2] identifies the sufficient flux and zero-slope forcing assumptions; [F7] fixes what stationary flow graph and maximality mean, while [F6] states exactly what parameter uniformity requires. These interfaces do not establish that a particular spacetime satisfies the quantitative bounds: the consumer must check differentiated background estimates, the common spacelike slope neighborhood, stationarity and the zero-slope forcing. The estimate just proved applies whenever these checks hold.

External input

In dimension $d$ , a scalar $z \in H^{1} (B_{1})$ weakly satisfying $div (B D z) = 0$ , with measurable symmetric $λ_{*} I \leq B \leq Λ_{*} I$ , obeys $∥ z ∥_{C^{0, θ} (B_{1 / 2})} \leq C ∥ z ∥_{L^{2} (B_{1})}$ for some $θ > 0$ , with $C, θ$ depending only on $d, λ_{*}, Λ_{*}$ . This is the symmetric real scalar case of Theorem 3.17 in Fernández-Real and Ros-Oton, Regularity Theory for Elliptic PDE, printed p. 93 (current PDF page 99). Only this homogeneous estimate is imported as [F20]; its proof is not reconstructed. The differentiated equation, lifting and restriction steps supply the local application under finite flux regularity.

Remarks

The original realized-coefficient estimate is retained. The added stationary-flux route supplies precise sufficient hypotheses for the intended graph application and separates preliminary nonlinear control from the final linear estimate. The original article is not cited as a prerequisite source. No claim about an error in that article is made.

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External dependencies

Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 3.17, printed p. 93, PDF page 99 (zero-based page 98 in the currently retrieved 236-page PDF)

In dimension $d$, a scalar $z\in H^1(B_1)$ weakly satisfying $\operatorname{div}(B Dz)=0$, with measurable symmetric $\lambda_* I\le B\le\Lambda_* I$, obeys $\|z\|_{C^{0,\theta}(B_{1/2})}\le C\|z\|_{L^2(B_1)}$ for some $\theta>0$, with $C,\theta$ depending only on $d,\lambda_*,\Lambda_*$.

Hypotheses: Real scalar weak solution on a Euclidean unit ball; bounded measurable uniformly elliptic symmetric matrix. Applied after translation and dilation in dimension d=n+1 to the lifted homogeneous equation, with no lower-order terms.

Reading limits: Read the complete extracted statement and immediate proof note of Theorem 3.17 on printed p. 93 in the author-hosted PDF during this assignment. The De Giorgi–Nash proof and its cited preceding results were not reconstructed. The lifting and finite-regularity application are local arguments here.

Sources

Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 3.17, printed p. 93, PDF page 99 (zero-based page 98 in the currently retrieved 236-page PDF)