Journal
ADVANCES IN CALCULUS OF VARIATIONS
Volume 16, Issue 4, Pages 975-1059Publisher
WALTER DE GRUYTER GMBH
DOI: 10.1515/acv-2021-0077
Keywords
Sub-elliptic partial differential equations; Kohn-Laplacian; Dirichlet problem; Neumann problem; Heisenberg group; singular integrals
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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.
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.
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