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The p-Laplacian is an admissible monotone diffusion

Statement

Assume the Axiom of Choice. Let $Ω \subset R^{d}$ be a nonempty bounded open set, $d \geq 1$ , and let $p > \max (1, 2 d / (d + 2))$ . The flux $a (x, ξ) = ∣ ξ ∣^{p - 2} ξ$ , with $a (x, 0) = 0$ , satisfies the coercivity, monotonicity and $p$ -growth hypotheses in Coercive monotone p-growth diffusion fluxes, and its induced operator is additionally norm-continuous. Its zero-boundary space embeds compactly into $L^{2} (Ω)$ . In particular, on the nonempty interval $Ω = (0, 1)$ for $d = 1$ , or the nonempty unit cube $Ω = (0, 1)^{d}$ for $d \geq 2$ , the explicit fixed-degree interval or Kuhn/Freudenthal meshes and any independent time partitions with vanishing maximum step satisfy the projector hypotheses. The convergence theorem Conditional monotone finite-element convergence therefore applies on those concrete meshes for every load and initial datum in its stated classes; no projector condition remains outstanding in this example.

Facts & Assumptions

Given: The Axiom of Choice, the stated $Ω, d, p$ , and the definitions of the flux and convergence hypotheses.

[F1]

An admissible flux is Carathéodory, has $p$ -growth and coercivity, and induces a bounded map $V \to V^{*}$ (Coercive monotone p-growth diffusion fluxes).

[F2]

For the stated strict exponent threshold, $W_{0}^{1, p} (Ω) ⋐ L^{2} (Ω)$ (The subcritical W0 one p embedding into L2).

[F3]

Under Choice, the nonlinear FEM convergence theorem gives subsequential strong $L^{r} (H)$ convergence and identifies the weak flux limit, conditional on the stable projector interface (Conditional monotone finite-element convergence, Conforming FEM families with stable L2 projectors).

[F4]

The fixed-degree uniform interval and cube meshes have $L^{2}$ projectors uniformly bounded on $W_{0}^{1, p}$ and strongly approximating there and in $L^{2}$ ; their time partitions may refine independently (L2 projections are uniformly W1p-stable on quasi-uniform conforming meshes).

[A1]

The Axiom of Choice is assumed by the concrete flux-space definition and its compact Sobolev embedding interface (The Axiom of Choice, Coercive monotone p-growth diffusion fluxes, The subcritical W0 one p embedding into L2).

Proof

1.1F1givenalgebra

Let $J_{p} (ξ) = ∣ ξ ∣^{p - 2} ξ$ and $Φ (ξ) = ∣ ξ ∣^{p} / p$ . The function $Φ$ is convex: the triangle inequality gives $∣ θ ξ + (1 - θ) η ∣ \leq θ ∣ ξ ∣ + (1 - θ) ∣ η ∣$ , and $r \mapsto r^{p}$ is increasing and convex for $r \geq 0$ . It is differentiable with $\nabla Φ = J_{p}$ , including at zero because $p > 1$ . For a differentiable convex function, restriction to the segment from $η$ to $ξ$ gives the tangent inequalities $Φ (ξ) \geq Φ (η) + J_{p} (η) \cdot (ξ - η), Φ (η) \geq Φ (ξ) + J_{p} (ξ) \cdot (η - ξ) .$ Adding them yields $(J_{p} (ξ) - J_{p} (η)) \cdot (ξ - η) \geq 0$ . Also $J_{p} (ξ) \cdot ξ = ∣ ξ ∣^{p}$ and $∣ J_{p} (ξ) ∣ = ∣ ξ ∣^{p - 1}$ , giving monotonicity, coercivity with $α = 1, h = 0$ , and growth with $C = 1, g = 0$ .

1.2F1givenalgebra

For $p \geq 2$ , the derivative of $J_{p}$ has operator norm at most $(p - 1) ∣ z ∣^{p - 2}$ . Integrating it along the segment from $η$ to $ξ$ gives $∣ J_{p} (ξ) - J_{p} (η) ∣ \leq (p - 1) (∣ ξ ∣ + ∣ η ∣)^{p - 2} ∣ ξ - η ∣ .$ For $1 < p < 2$ , put $d_{0} = ∣ ξ - η ∣$ and $M_{0} = \max (∣ ξ ∣, ∣ η ∣)$ . If $d_{0} \geq M_{0} / 2$ , then $∣ J_{p} (ξ) - J_{p} (η) ∣ \leq 2 M_{0}^{p - 1} \leq 2^{p} d_{0}^{p - 1}$ . If $0 < d_{0} < M_{0} / 2$ , every point on the segment has norm at least $M_{0} / 2$ ; there the derivative norm is at most $(M_{0} / 2)^{p - 2}$ . Hence $∣ J_{p} (ξ) - J_{p} (η) ∣ \leq (M_{0} / 2)^{p - 2} d_{0} \leq C_{p} d_{0}^{p - 1},$ because $(d_{0} / M_{0})^{2 - p} \leq 1$ . The case $d_{0} = 0$ is immediate. Thus for $1 < p < 2$ the global estimate $∣ J_{p} (ξ) - J_{p} (η) ∣ \leq C_{p} ∣ ξ - η ∣^{p - 1}$ holds.

2.1F1step 1.2givenalgebra

If $z_{n} \to z$ in $V$ , then $\nabla z_{n} \to \nabla z$ in $L^{p}$ . For $p \geq 2$ , Hölder applied to the first estimate of step 1.2 gives $∥ J_{p} (\nabla z_{n}) - J_{p} (\nabla z) ∥_{L^{p^{'}}} \leq (p - 1) (∥ \nabla z_{n} ∥_{p} + ∥ \nabla z ∥_{p})^{p - 2} ∥ \nabla z_{n} - \nabla z ∥_{p} \to 0.$ For $1 < p < 2$ , the second estimate and $(p - 1) p^{'} = p$ give $∥ J_{p} (\nabla z_{n}) - J_{p} (\nabla z) ∥_{L^{p^{'}}} \leq C_{p} ∥ \nabla z_{n} - \nabla z ∥_{p}^{p - 1} \to 0.$ For any $w \in V$ , Hölder bounds $∣ ⟨ A (z_{n}) - A (z), w ⟩ ∣ \leq ∥ J_{p} (\nabla z_{n}) - J_{p} (\nabla z) ∥_{p^{'}} ∥ w ∥_{V}$ . Therefore $A : V \to V^{*}$ is norm-continuous, an additional property of this example.

3.1A1F1F2step 1.1step 2.1given

The threshold and [F2] supply the required dense compact $V ⋐ L^{2} (Ω)$ inclusion, and step 1.1 verifies the pointwise flux conditions. Step 2.1 additionally proves norm continuity, without requiring it as a flux hypothesis. Hence the flux meets all hypotheses of [F1].

4.1F3F4step 3.1given∎

On $Ω = (0, 1)$ in $d = 1$ the threshold is $p > 1$ , while on the unit cube in $d \geq 2$ it is exactly the stated strict threshold. For either domain, [F4] supplies an explicit fixed-degree spatial family and time partitions satisfying the definition in [F3]. Applying the convergence theorem [F3] to the flux verified in step 3.1 gives the concrete subsequential FEM convergence assertion in the Statement, with all original data and exponent hypotheses retained.

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