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The Dirichlet Problem for the Subelliptic Laplacian on the Heisenberg Group II

Published online by Cambridge University Press:  20 November 2018

Bernard Gaveau  and
Jacques Vauthier
Bernard Gaveau
Affiliation:
Université Pierre et Marie Curie, Paris, France
Jacques Vauthier
Affiliation:
Université Pierre et Marie Curie, Paris, France
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Let H3 be the Heisenberg group in three dimensions, Δ the fundamental subelliptic laplacian on H3 (see Section 1 for notations and definitions) and U be an open subset of H3 If φ is a continuous function on the boundary ∂U of U, the Dirichlet problem is thus,

(1)

In [3], p. 104, it was asserted by the first author that, when dU is regular (see Section 1 for this definition), the problem (1) has a solution continuous on D and a probabilistic formula was given. In [3], we prove that our probabilistic formula gives a solution of the so called “martingale problem” associated to Δ on U (see [5] for this notion). But it appears that the connection between the solution in the martingale problem sense and the true solution is not at all clear in the subelliptic case; in particular it is not obvious at all that the probabilistic formula is a C2 function.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 760 - 766
Copyright
Copyright © Canadian Mathematical Society 1985

References

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