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 −Lu=f(u), where L is a linear uniformly elliptic operator and f is C1, such that the linearized equation −L−f′(u) has nonnegative principal eigenvalue. Our main result is an estimate for the L2+γ 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 C1,1 domains if we further assume the nonlinearity f to be convex. This result is new even for the Laplacian, for which a C3 regularity assumption on the domain was needed.