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

Takashi Suzuki, Takuya Tsuchiya

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.

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

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.
Paper Structure (14 sections, 23 theorems, 277 equations)

This paper contains 14 sections, 23 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poisson equation on Lipschitz domains
  4. Differentiations on $\partial\Omega$
  5. Domain perturbations
  6. Jacobi matrix and its derivatives
  7. Liouville's formulae
  8. First formulae
  9. Second formulae
  10. Hadamard variation of the Green's function
  11. First variation
  12. Second variation
  13. Liouville's area formula and differential forms
  14. Second fundamental form on $\partial\Omega$

Key Result

Theorem 1

If $\Omega\subset \mathbb R^d$ is a Lipschitz domain, then the set of functions $C^\infty(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$ for $1\leq p<\infty$, where

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • Theorem 7: first volume formula
  • Remark 8
  • Remark 9
  • ...and 35 more