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 $-\Delta u = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $\Omega \subset \mathbb{R}^n$ 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 $\Omega$ or, equivalently, if $-\Delta u = f(u)<+\infty$ in $\Omega$. 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.