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

We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $Ω\subset\mathbb{R}^3$ 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 $Du(Ω)$. We