Sub-elliptic boundary value problems in flag domains

A flag domain in R-3 is a subset of R-3 of the form {(x, y, t) : y < A(x)}, where A: R -> R is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn Laplacian Delta(b) = X-2 + Y-2 in flag domains Omega subset of 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.

Automated summary

This paper investigates the sub-elliptic problems in flag domains and their boundaries in R-3. By utilizing the sub-elliptic single and double layer potentials, we solve the Dirichlet and Neumann problems and obtain improved regularity for the solutions with certain regularity of the boundary values.