Existence of Solutions to the Second Boundary-Value Problem for the $p$-Laplacian on Riemannian Manifolds

V. V. Brovkin ; A. A. Kon'kov

Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 180-195

Résumé

We obtain necessary and sufficient conditions for the existence of solutions to the boundary-value problem $$ \Delta_p u=f\quad\text{on}\quad M,\qquad |\nabla u|^{p-2}\,\frac {\partial u}{\partial \nu}\bigg|_{\partial M}=h, $$ where $p > 1$ is a real number, $M$ is a connected oriented complete Riemannian manifold with boundary, and $\nu$ is the outer normal vector to $\partial M$.

Détail

Keywords: $p$-Laplacian, Riemannian manifold, Dirichlet integral.