Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶^1,1 domains

Abstract

This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions

, for

C 1 , 1 C^{1\smash{,}1}

domains. We consider equations

− L ⁢ u = f ⁢ ( u ) -Lu=f(u)

in a bounded

C 1 , 1 C^{1\smash{,}1}

domain

Ω ⊂ R n \Omega\subset\mathbb{R}^{n}

, with

u = 0 u=0

on

∂ Ω \partial\Omega

, where 𝐿 is a linear elliptic operator with variable coefficients and

f ∈ C 1 f\in C^{1}

is nonnegative, nondecreasing, and convex. The stability of 𝑢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation

− L − f ′ ⁢ ( u ) -L-f^{\prime}(u)

. Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), 2, 187–252] proved the Hölder continuity in

C 3 C^{3}

domains.