On the Laplace equation on bounded subanalytic manifolds

TL;DR

This paper develops a trace formula and density results for tangent vector fields on bounded subanalytic manifolds, leading to new existence and uniqueness theorems for Laplace and p-Laplace equations with boundary conditions.

Contribution

It introduces a trace formula and density results for tangent vector fields on subanalytic manifolds, enabling new solutions for Laplace equations with boundary conditions.

Findings

Established a trace formula for integration by parts on subanalytic manifolds.

Proved density of certain Sobolev spaces of tangent vector fields.

Derived existence and uniqueness results for Laplace and p-Laplace equations.

Abstract

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of $\mathbb{R}^n$, possibly non closed. We also establish density results for $\mathbf{W}^{1,p}_\nabla (M)$, $M$ bounded subanalytic manifold, which is the space of the $L^p$ tangent vector fields $v$ on $M$ for which $\nabla v$ is $L^p$, where $\nabla$ is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the $p$-Laplace equation, for $p\in [1,\infty)$ large.