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Regularity of stable solutions to the MEMS problem up to the optimal dimension 6

Renzo Bruera, Xavier Cabre

Abstract

In this article we address the regularity of stable solutions to semilinear elliptic equations $-Δu = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $Ω\subset \mathbb{R}^n$ and $f:[0,1)\to (0,+\infty)$ blowing up at $u=1$ and nonintegrable near 1. In this context, a solution $u$ is regular if $u<1$ in all $Ω$ or, equivalently, if $-Δu = f(u)<+\infty$ in $Ω$. This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that $u$ may satisfy. Our results hold up to the optimal dimension $n=6$ (there are counterexamples for $n\geq 7$) but require a Crandall-Rabinowitz type assumption on the nonlinearity $f$. Our main estimate controls the $L^\infty$ norm of $F(u)$ in a ball, where $F$ is a primitive of $f$, by only the $L^1$ norm of $u$ in a larger ball. Under the same assumptions, we also give global estimates in dimensions $n\leq 6$ for the Dirichlet problem with vanishing boundary condition, improving previously known results. For $n\leq 2$, we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1.

Regularity of stable solutions to the MEMS problem up to the optimal dimension 6

Abstract

In this article we address the regularity of stable solutions to semilinear elliptic equations with MEMS type nonlinearities. More precisely, we will have in a domain and blowing up at and nonintegrable near 1. In this context, a solution is regular if in all or, equivalently, if in . This paper establishes for the first time interior regularity estimates that are independent of the boundary condition that may satisfy. Our results hold up to the optimal dimension (there are counterexamples for ) but require a Crandall-Rabinowitz type assumption on the nonlinearity . Our main estimate controls the norm of in a ball, where is a primitive of , by only the norm of in a larger ball. Under the same assumptions, we also give global estimates in dimensions for the Dirichlet problem with vanishing boundary condition, improving previously known results. For , we do not need a Crandall-Rabinowitz type assumption and, thus, our global estimate holds for all nonnegative, nondecreasing, convex nonlinearities which blow up at 1 and are nonintegrable near 1.

Paper Structure

This paper contains 12 sections, 10 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Overview of the setting and results
  3. Explicit stable singular solutions
  4. Main results
  5. Structure of the proofs
  6. Interior estimates through powers of the Laplacian
  7. Interior estimates through powers of $\abs{x}$
  8. Estimates up to the boundary
  9. On the value of the Crandall-Rabinowitz constant
  10. On the value of $\underline{\gamma}$ for regular and singular nonlinearities
  11. On the relationship between the constants $\gamma$, $m$, and $q$
  12. On the classes of nonlinearities with $\underline{\gamma}>1$ or $q>0$

Key Result

Theorem 1.1

Let $u\in C^2(\overline{B_1})$ be a nonnegative stable solution of $-\Delta u = f(u)$ in $B_{1}\subset \mathbb{R}^n$ for some function $f\in C^2([0,1))$ satisfying $f\geq 0$, $f'\geq 0$, $F(1)=+\infty$ (where $F'=f$), and that $f^{-\theta}$ is concave near 1 for some $\theta>0$. In particular, we ha for some constant $C$ depending only on $n$, $\theta$, and $K$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Lemma 1.7
  • proof
  • proof : Proof of Proposition \ref{['proposition_first_proof']}
  • Lemma 2.1
  • ...and 14 more