Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem

TL;DR

This paper extends regularity results for a class of infinitely degenerate quasilinear equations, showing solutions are smooth under weaker bounds and establishing well-posedness for the Dirichlet problem.

Contribution

It introduces new bounds based solely on the supremum norm for a subclass of equations and proves smoothness and well-posedness of solutions.

Findings

Solutions are smooth if continuous weak solutions are considered.

Existence and uniqueness of solutions for the Dirichlet problem are established.

Interior regularity of solutions is proven.

Abstract

In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.