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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>9</m:mn> </m:mrow> </m:math> n\leq 9 , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> C^{1\smash{,}1} domains. We consider equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> -Lu=f(u) in a bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> C^{1\smash{,}1} domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> \Omega\subset\mathbb{R}^{n} , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> u=0 on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0em">∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> \partial\Omega , where 𝐿 is a linear elliptic operator with variable coefficients and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>L</m:mi> </m:mrow> <m:mo>−</m:mo> <m:mrow> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> C^{3} domains.