Geometric gradient estimates for solutions to degenerate elliptic equations

Abstract

We establish new, optimal geometric gradient estimates for solutions to a class of 2nd order partial differential equations, \(\mathcal {L}(X, \nabla u, D^2 u) = f\), whose diffusion properties (ellipticity) degenerate along the a priori unknown set of critical points of an existing solution, \(\mathcal {S}(u) := \{ X : \nabla u(X) = 0 \}\). Such a quantitative information plays a decisive role in the analysis of a number of analytic and geometric problems. The results proven in this work are new even for simple equation \(|\nabla u | \cdot \Delta u = 1\). Under natural nondegeneracy condition on the source term \(f\), we further establish the precise asymptotic behavior of solutions at interior gradient vanishing points. These new estimates are then employed in the study of some well known problems in the theory of elliptic PDEs.