Abstract
We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc., as well as in a strip.
Highlights
Prototype H-type groups are an important class of homogeneous stratified Lie groups of step two since any H-type group is naturally isomorphic to a prototype H-type group
The main aim of this short note is to give an answer to the question: Is there a class of domains in which the Dirichlet boundary value problem (1.1) is explicitly solvable in the classical sense?
Kac’s question: Is there any boundary value problem for the Laplacian which is explicitly solvable in the classical sense for any smooth domain?
Summary
Prototype H-type groups are an important class of homogeneous stratified Lie groups of step two since any (abstract) H-type group is naturally isomorphic to a prototype H-type group. The main aim of this short note is to give an answer to the question: Is there a class of domains in which the Dirichlet boundary value problem (1.1) is explicitly solvable in the classical sense? Kac’s question: Is there any boundary value problem for the Laplacian which is explicitly solvable in the classical sense for any smooth domain?. It is interesting to note that the explicit solutions in these papers have been constructed for Kac’s boundary value problem for the sub-Laplacian well in the presence of characteristic points on the boundary.
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