A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

We study stable solutions to the equation (−Δ)1/2u=f(u), posed in a bounded domain of Rn. For nonnegative convex nonlinearities, we prove that are smooth dimensions n≤4. This result, which was known only for n=1, follows from new interior Hölder estimate is completely independent nonlinearity f. A main ingredient our proof geometric form stability condition. It still unknown other fractions Laplacian and, surprisingly, it requires convexity nonlinearity. From it, deduce higher order Sobolev estimates allow us extend techniques developed by Cabré, Figalli, Ros-Oton, and Serra Laplacian. In this way obtain, besides bound n≤4, universal H1/2 all dimensions. Our L∞ expected hold n≤8, but has been settled radial case or when f(u)=λeu. Laplacian, optimal dimension boundedness reached f(u)=λeu, even case.