A universal H\"older estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

TL;DR

This paper establishes a universal interior H"older estimate for stable solutions to half-Laplacian semilinear equations in dimensions up to 4, independent of the nonlinearity, and extends regularity results using new geometric stability conditions.

Contribution

The paper introduces a new geometric form of the stability condition for half-Laplacian equations, enabling the proof of smoothness of solutions in dimensions up to 4 for convex nonlinearities.

Findings

Stable solutions are smooth in dimensions n ≤ 4.

A new interior H"older estimate independent of the nonlinearity.

Universal H^{1/2} estimate in all dimensions.

Abstract

We study stable solutions to the equation $(-\Delta)^{1/2} u = f(u)$, posed in a bounded domain of $\mathbb{R}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\leq 4$. This result, which was known only for $n=1$, follows from a new interior H\"older estimate that is completely independent of the nonlinearity $f$. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabr\'e, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the H\"older bound for $n\leq 4$, a universal $H^{1/2}$ estimate in all dimensions. Our $L^\infty$ bound is expected to hold for $n\leq 8$, but this has been settled only in the radial case or when $f(u) = \lambda e^u$. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when $f(u) = \lambda e^u$, even in the radial case.