Optimal gradient continuity for degenerate elliptic equations

Abstract

We establish new, optimal gradient continuity 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 \textit{a priori} unknown singular set of an existing solution, $\mathcal{S}(u):=\{X:\nabla u(X)=0\}$. The innovative feature of our main result concerns its optimality -- the sharp, encoded smoothness aftereffects of the operator. Such a quantitative information usually plays a decisive role in the analysis of a number of analytic and geometric problems. Our result is new even for the classical equation $|\nabla u|\cdot \Delta u=1$. We further apply these new estimates in the study of some well known problems in the theory of elliptic PDEs.

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References in corpus (2)

• Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations
• Nonclassical Solutions of Fully Nonlinear Elliptic Equations II. Hessian Equations and Octonions