Optimal gradient continuity for degenerate elliptic equations

TL;DR

This paper derives optimal gradient continuity estimates for solutions to degenerate elliptic PDEs, especially near singular sets where the ellipticity degenerates, providing sharp regularity results with broad applications.

Contribution

It introduces the first optimal gradient continuity estimates for degenerate elliptic equations with unknown singular sets, advancing the understanding of solution regularity.

Findings

Established sharp gradient continuity estimates near singular sets.

Applied estimates to classical and well-known elliptic PDE problems.

Demonstrated the optimality of regularity results in degenerate elliptic equations.

Abstract

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nabla u(X) = 0 \}$. 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.