The Poincaré inequality and $p$-connectedness of a stratified set

We extend the Poincaré inequality to functions of Sobolev type on a stratified set. The integrability exponents in these analogs depend on the geometric characteristics of the stratified set which show to what extent their strata are connected with each other and the boundary. We apply the results to proving the solvability of boundary value problems for the $p$-Laplacian with boundary conditions of Neumann or Wentzel type.