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

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 $n \leq 9$.