Well-Posed Boundary Value Problems for the Laplace-Beltrami Operator on a Stratified Set
Keywords:
Laplace-Beltrami operator, correct restriction, well-posed problem, stratified set, Green function, regularized trace, self-adjoint restrictionAbstract
Correct restrictions of the shifted Laplace-Beltrami operator on a stratified set consisting of two punctured two-dimensional spheres connected by one segment are studied. Boundary functionals are constructed by combining regularized point traces at the punctures of the spherical strata with endpoint values and oriented derivatives on the connecting interval. A Green formula on the stratified set is obtained. Using this formula, restrictions of the maximal operator generated by linear boundary conditions
AuBu GG 01 += 0
are described. A criterion of correct solvability is reduced to the invertibility of the finite-dimensional matrix A + BM (l). Self-adjoint restrictions are characterized by the condition
AB* = BA*
under the rank condition
rank[A B] = 4.
A natural continuity-type coupling of the two spherical strata through the segment is presented as an example.
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