Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators

TL;DR

This paper explores the link between Orlicz-Sobolev inequalities and the solvability of the Dirichlet problem for infinitely degenerate elliptic operators, aiming to refine conditions for solvability in complex degenerate cases.

Contribution

It advances understanding by narrowing the gap between necessary and sufficient conditions for Dirichlet problem solvability in infinite degeneracy scenarios.

Findings

Established a stronger connection between Orlicz-Sobolev inequalities and Dirichlet problem solvability.

Reduced the gap between necessary and sufficient conditions for infinitely degenerate elliptic operators.

Extended classical results from subelliptic to infinitely degenerate cases.

Abstract

We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem.