Uniqueness of solutions to elliptic and parabolic equations on metric graphs

TL;DR

The paper extends uniqueness results for elliptic and parabolic PDEs to the setting of infinite metric graphs with Kirchhoff Laplacians. It develops weighted $L^p$-frameworks, using exponential and polynomial weights tied to the graph distance, and treats potentials and densities that are either bounded away from zero or decay at infinity. The main contribution is proving that, under appropriate growth conditions and, when necessary, a graph degree assumption for the tough $1\le p<2$ regime, the only solutions in the prescribed weighted spaces are the trivial ones for both elliptic and parabolic problems. This work broadens the understanding of diffusion and Schrödinger-type dynamics on networked structures and provides sharp uniqueness criteria in terms of spectral and geometric graph data.

Abstract

We investigate uniqueness of solutions to certain classes of elliptic and parabolic equations posed on metric graphs. In particular, we address the linear Schrödinger equation with a potential, and the heat equation with a variable density. We assume suitable growth conditions on the solutions, which are related to the behaviour at infinity of the potential or of the density.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Theorem 3.4
• Theorem 3.5
• Remark 3.6