Liouville's formulae and Hadamard variation with respect to general domain perturbations

TL;DR

This paper explores how domain perturbations affect geometric and boundary properties using Liouville's formulae, focusing on Hadamard variations and their relation to boundary geometry and differential forms.

Contribution

It introduces new Liouville's formulae for volume and area transformations under domain perturbations, linking them to boundary geometry and second fundamental form.

Findings

Derived new Liouville's formulae for domain transformations.

Established relations between Hadamard variations and boundary geometric quantities.

Analyzed implications for Neumann boundary conditions.

Abstract

We study Hadamard variations with respect to general domain perturbations, particularly for the Neumann boundary condition. They are derived from new Liouville's formulae concerning the transformation of volume and area integrals. Then, relations to several geometric quantities are discussed; differential forms and the second fundamental form on the boundary.