Energy estimate up to the boundary for stable solutions to semilinear elliptic problems

Abstract

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to $- L u = f(u)$, where $L$ is a linear uniformly elliptic operator and $f$ is $C^1$, such that the linearized equation $-L - f'(u)$ has nonnegative principal eigenvalue. Our main result is an estimate for the $L^{2+\gamma}$ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing $f$. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general $C^{1,1}$ domains if we further assume the nonlinearity $f$ to be convex. This result is new even for the Laplacian, for which a $C^3$ regularity assumption on the domain was needed.