Boundary value problem, associated with diffeomorphism between Riemannian manifolds
DOI:
https://doi.org/10.20535/SRIT.2308-8893.2018.1.11Keywords:
Hilbert space, Riemannian manifold, diffeomorphism, Borel measure, derivation of measures, Laplacian, Dirichlet problemAbstract
Laplace operator construction is considered in L2-version with respect to the measure in the context of diffeomorphism between (infinite-dimensional) Riemannian manifolds. The connection between such operators as the gradient closure, boundary restriction operator and divergence with respect to the measure on diffeomorphic manifolds is derived. It is proved that in the case when the gradient closure, boundary restriction operator and divergence with respect to measure are correctly defined on a Riemannian manifold, the respective operators are correctly defined on a diffeomorphic Riemannian manifold too. As a corollary of the derived connection, the class of solvable boundary value problems (problems that have one and only one solution) on Riemannian manifolds (and on Hilbert’s space as a particular case of Riemannian manifold) is widened by reducing the problem of a special kind into an associated with it Dirichlet problem.References
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