Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶1,1 domains

Abstract

Abstract This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 n\leq 9 , 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.