Weak Solvability of the Dirichlet Problem on Stratified Sets

The Dirichlet problem is posed for an analog of the Beltrami–Laplace operator on sets consisting of manifolds of various dimensions regularly adjacent to one another (stratified sets). A special system of notions permits one to prove analogs of Green's integral identities and the Poincaré inequality for Sobolev type spaces. The weak solvability of the Dirichlet problem for this operator, as well as for an analog of the biharmonic operator, is proved on the basis of the Riesz theorem on the representation of linear functionals.