Higher order operators on networks: hyperbolic and parabolic theory

TL;DR

This paper develops a variational framework for higher-order elliptic operators on networks, exploring hyperbolic and parabolic equations, their well-posedness, energy conservation, and introduces a new class of evolution equations with Wentzell boundary conditions.

Contribution

It extends second-order network operator theory to fourth-order cases, revealing new well-posed equations with Wentzell boundary conditions and analyzing their properties.

Findings

Extension of variational framework to fourth-order operators

Discovery of well-posed evolution equations with Wentzell boundary conditions

Analysis of regularizing properties of polyharmonic heat kernels

Abstract

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions.