Continuity of infinitely degenerate weak solutions via the trace method

TL;DR

This paper extends the continuity results for weak solutions of infinitely degenerate elliptic equations in the plane, removing previous geometric constraints under certain doubling conditions.

Contribution

It removes geometric constraints previously required in planar hypoelliptic equations with degenerate coefficients, under doubling conditions on F.

Findings

Continuity of weak solutions established without geometric constraints

Results apply to homogeneous equations with doubling F

Extends classical hypoelliptic theory to degenerate elliptic equations

Abstract

In 1971 Fedi\u{i} proved the remarkable theorem that the linear second order partial differential operator in the plane with coefficients 1 and f^2 is hypoelliptic provided that f is smooth, vanishes at the origin and is positive otherwise. Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, to infinitely degenerate elliptic divergence form equations where the nonnegative matrix A(x,u) has bounded measurable coefficients with trace roughly 1 and determinant comparable to f, and where F=ln(1/f) is essentially doubling. However, in the plane, these variants assumed additional geometric constraints on f, something not required in Fedi\u{i}'s theorem. In this paper we in particular remove these additional geometric constraints in the plane for homogeneous equations with F essentially doubling.