Well-Posedness of the Hodge Wave Equation on a Compact Manifold

TL;DR

This paper establishes well-posed boundary conditions for the Hodge wave equation on compact manifolds using differential geometry and boundary triplet theory.

Contribution

It introduces a boundary triplet framework for the Hodge wave equation and identifies conditions ensuring well-posedness on compact manifolds.

Findings

Boundary triplet for Hodge wave equation identified

Boundary conditions ensuring well-posedness determined

Sobolev spaces of differential forms characterized

Abstract

In this work, we study the Hodge wave equation on a compact orientable manifold. We present the necessary differential geometry language to treat Sobolev spaces of differential forms and use these tools to identify a boundary triplet for the problem. We use this boundary triplet to determine a class of boundary conditions for which the problem is well-posed.