Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients

TL;DR

This paper establishes an existence theory for weak solutions to certain second order linear subelliptic Dirichlet problems with rough coefficients, using advanced functional analysis techniques in degenerate Sobolev spaces.

Contribution

It introduces a new existence framework for weak solutions of non-elliptic PDEs with degenerate principal parts, extending previous regularity results.

Findings

Existence of weak solutions under degenerate conditions.

Application of degenerate Sobolev spaces to non-elliptic PDEs.

Extension of regularity theory to broader class of equations.

Abstract

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}\Theta u&=&\phi\textrm{on}\partial \Theta.{eqnarray} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\cal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Omega,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in recent work of E. Sawyer and R. L. Wheeden. The aforementioned authors in referenced work give a regularity theory for a subset of the class of equations dealt with here.