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Comments on the quantitative uniqueness of continuation for evolution equations

Mourad Choulli

Abstract

We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy estimates.

Comments on the quantitative uniqueness of continuation for evolution equations

Abstract

We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy estimates.
Paper Structure (3 sections, 9 theorems, 61 equations)

This paper contains 3 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['MT1']}, \ref{['MT2']} and \ref{['MT3']}
  3. Energy estimates

Key Result

Theorem 1.1

Assume that $\Omega$ is $C^{1,1}$ or $\Omega$ is convex. Let $0\le \psi \in C^4(\overline{\Omega})$ such that $\mathfrak{m}_\psi>0$. For any $\lambda \ge \lambda_1$ and $u\in \mathcal{H}_0(Q)$ satisfying $Hu=0$ we have Furthermore, if $u(\cdot ,0)\in H^\theta(\Omega)$, for some $\theta >1$, then

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.2
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['MT1']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['MT2']}
  • Theorem 2.3
  • ...and 6 more