Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pointwise Hadamard variational formula for the fractional Laplacian

Sidy M. Djitte, Franck Sueur

Abstract

We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-Δ)^s u = h$ in $Ω$ and $u=0$ in $Ω^c$, where the right-hand side $h$ is either a Dirac delta distribution or a Lipschitz function. In both cases, we prove that the corresponding solution is shape differentiable in every direction and we derive a formula for the pointwise value of its shape derivative. These formulas involve integral on the domain's boundary and fractional Neumann's traces. This extends to the case of the fractional Laplacian the well-known Hadamard variational formula for the standard Laplacian. Our argument is in the spirit of \cite{Ushikoshi, Kozono-Ushikoshi} and is based on PDEs techniques.

Pointwise Hadamard variational formula for the fractional Laplacian

Abstract

We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation in and in , where the right-hand side is either a Dirac delta distribution or a Lipschitz function. In both cases, we prove that the corresponding solution is shape differentiable in every direction and we derive a formula for the pointwise value of its shape derivative. These formulas involve integral on the domain's boundary and fractional Neumann's traces. This extends to the case of the fractional Laplacian the well-known Hadamard variational formula for the standard Laplacian. Our argument is in the spirit of \cite{Ushikoshi, Kozono-Ushikoshi} and is based on PDEs techniques.
Paper Structure (15 sections, 25 theorems, 449 equations)

This paper contains 15 sections, 25 theorems, 449 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Notations
  3. Preliminary estimates
  4. A useful regularity estimate
  5. Some convergence results
  6. Differentiability of the mapping $t\mapsto H^s_{\Omega_t}(\Phi_t(x),\Phi_t(y))$
  7. First step toward the derivation of the formula \ref{['var-green']}
  8. Derivation of the formula \ref{['var-green']} and Proof of Theorem \ref{['THM-1.5']}
  9. Proof of Corollary \ref{['cor6']}
  10. Proof of Theorem \ref{['THM-1.2']}
  11. Proof of Lemma \ref{['Useful-bounds']}
  12. Proof of Lemma \ref{['FundLemma2']}
  13. A useful integral identity involving the kernel $\omega^x_{Y}(\cdot,\cdot)$
  14. Estimates on the cut-off function $\xi_k$
  15. Estimates on the Green function

Key Result

Theorem 1.2

Let $\Omega\subset\mathbb{R}^N$ be a bounded open set of class $C^{1,1}$ with $N>2s$ and let $s\in (0,1)$. Let $h$ be Lipschitz continuous in $D$, where $\Omega\Subset D\subset \mathbb{R}^N$ and $u$ be the unique weak solution of Dir-01. Let $Y$ be a globally $C^{1,1}$ vector field in $\mathbb{R}^N$ In the above, $\Gamma$ stands for the classical gamma function and $\nu$ denotes the outward unit n

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 42 more