A geometric approach to nonlocal 2-Hessian equations

Abstract: We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $$\inf_{A\in {A}_2}Δ^s (u\circ A)(A<sup>{-1}x)$$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.