Convex Functions over the Whole Space Locally Satisfying Fractional Equations

Abstract

We investigate the structure of convex functions over the whole space which satisfy in some convex domain an equation involving the fractional Laplacian. Roughly speaking, it turns out that such solutions are either strictly convex in the given domain, or degenerate in the sense that their graph is a ruled hypersurface. We also consider regular solutions, that some fractional equations admit, and show that the convexity of the datum is transmitted to the solution through its regularity. The results are obtained by means of a fractional form of the celebrated convexity maximum principle devised by Korevaar in the 80’s. More precisely, we construct an anisotropic, degenerate, fractional operator that nevertheless satisfies a maximum principle, and we apply such an operator to the concavity function associated to the solution. An explicit, two-dimensional example is also constructed.