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We establish two new estimates which control a function in $L^1$ by only the$L^1$ norm of its radial derivative. While the interior estimate holds for allsuperharmonic functions, the boundary version is much more delicate. Itrequires the function to be a stable solution of a semilinear elliptic equationwith a nonnegative, nondecreasing, and convex nonlinearity. As an application,our estimates provide quantitative proofs of two results established in[Cabr\'e, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] bycontradiction-compactness proofs. We recall that this work proved the H\"olderregularity of stable solutions to semilinear elliptic equations in the optimalrange of dimensions $n\leq 9$.

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