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New quantitative uniqueness of continuation for elliptic equations

Mourad Choulli, Hiroshi Takase

Abstract

We prove a new quantitative uniqueness of continuation for elliptic equations from Cauchy data. We provide a simple and direct proof based only on a Carleman inequality. Similar result for the Stokes equation is also shown.

New quantitative uniqueness of continuation for elliptic equations

Abstract

We prove a new quantitative uniqueness of continuation for elliptic equations from Cauchy data. We provide a simple and direct proof based only on a Carleman inequality. Similar result for the Stokes equation is also shown.

Paper Structure

This paper contains 3 sections, 5 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['interpolation_inequality']}
  3. Extension to the Stokes equation

Key Result

Theorem 1.1

Let $0\le\eta<2$ and $\zeta:=(g,X,p,B,\Omega,\eta)$. Then there exist $\mathbf{c}=\mathbf{c}(\zeta)>0$ and $c=c(\zeta) >0$ such that for any $u\in H^2(D)$ and $s\ge 1$ we have

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof : Proof of Theorem \ref{['interpolation_inequality']}
  • Theorem 3.1
  • Corollary 3.2
  • proof : Proof of Theorem \ref{['interpolation_Stokes']}