An analog of the Sobolev inequality on a stratified set

Abstract:

On a set $\Omega \subset \mathbb {R}^d$ that is formed from a finite collection of manifolds that adjoin each other in a special way (i.e., on a stratified set) an exact analog of the Sobolev inequality is proved, in which the exponents are expressed via the characteristics of the intrinsic geometric structure of $\Omega$. The result is applied in order to prove the solvability of the Dirichlet problem for the $p$-Laplacian.