Lewy-Type Nondegeneracy for Gradient-Dependent Elliptic Equations and Global Gradient Diffeomorphisms on Convex Rings

Abstract

We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $\Omega\subset\mathbb{R}^3$ and let $u$ solve \[ a^{ij}(Du)u_{ij}=0, \] where $a$ is a smooth, symmetric, positive definite matrix field on an open set containing $Du(\Omega)$. We show that if the gradient map $Du$ is a local homeomorphism, then $\det D^2u$ never vanishes; hence $Du$ is a local $C^\infty$-diffeomorphism. Second, in every dimension, we consider capacitary solutions on convex rings. If the solution has no critical points and its level hypersurfaces are strictly convex, then ellipticity alone forces the Hessian to have one positive and $n-1$ negative eigenvalues, that is, inertia $(1,n-1)$. Consequently, the gradient is a global diffeomorphism onto a radially parametrized ring in gradient space. Both results apply to the $p$-Laplace and minimal surface equations. For the global minimal-surface result, existence of a smooth solution is assumed. Classical convex-ring results supply the noncriticality and strict level-set convexity needed in the global corollaries. Since the two coefficient matrices are real analytic on the relevant gradient ranges, the corresponding local and global gradient diffeomorphisms are real analytic.