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arXiv · 2211.13033

Estimates controlling a function by only its radial derivative and applications to stable solutions of elliptic equations

Xavier Cabre
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Abstract

We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is much more delicate. It requires the function to be a stable solution of a semilinear elliptic equation with a nonnegative, nondecreasing, and convex nonlinearity. As an application, our estimates provide quantitative proofs of two results established by contradiction-compactness arguments in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)]. We recall that this work proved the H\"older regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$.

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Xavier Cabre. 2022-11-23. Estimates controlling a function by only its radial derivative and applications to stable solutions of elliptic equations. https://arxiv.org/abs/2211.13033

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