Sub-elliptic boundary value problems in flag domains

TL;DR

This paper addresses boundary value problems for the sub-elliptic Kohn-Laplacian in flag domains, introducing layer potential methods on the Heisenberg group to establish existence, regularity, and boundary behavior of solutions.

Contribution

It develops a comprehensive theory of layer potentials for sub-elliptic operators on flag domains, solving Dirichlet and Neumann problems with improved regularity results.

Findings

Solutions constructed via layer potentials as integral operators.

Existence of solutions with $L^{2}$ boundary data.

Enhanced regularity for solutions with Sobolev boundary data.

Abstract

A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn-Laplacian $\bigtriangleup^{\flat} = X^{2} + Y^{2}$ in flag domains $\Omega \subset \mathbb{R}^{3}$, with $L^{2}$-boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order $L^{2}$-Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.