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The property of unique continuation for second order evolution PDEs

Mourad Choulli

Abstract

We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and Schödinger operators with time-independent principal part. Our method is builds on two-parameter Carleman inequalities combined with unique continuation across a pseudo-convex hypersurface with respect to the space variable. The most results we demonstrate in this work are more or less classical. Some of them are not stated exactly as in their original form.

The property of unique continuation for second order evolution PDEs

Abstract

We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and Schödinger operators with time-independent principal part. Our method is builds on two-parameter Carleman inequalities combined with unique continuation across a pseudo-convex hypersurface with respect to the space variable. The most results we demonstrate in this work are more or less classical. Some of them are not stated exactly as in their original form.

Paper Structure

This paper contains 22 sections, 35 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main notations and assumptions
  4. Pseudo-convexity condition
  5. Carleman weights
  6. Pseudo-convex hypersurface
  7. The wave equation
  8. Carleman inequality
  9. Geometric form of the Carleman inequality
  10. Unique continuation
  11. Observability inequality
  12. Elliptic equations
  13. Carleman inequality
  14. Unique continuation
  15. Parabolic equations
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $A\in \mathscr{M}(\Omega ,\varkappa,\mathfrak{m})$ and $\mathcal{O}$ an orthogonal transformation. Then $A_{\mathcal{O}}\in \mathscr{M}(\mathcal{O}\Omega,\varkappa ,\mathfrak{m}')$, where $\mathfrak{m}'=\mathfrak{m}'(n,\mathfrak{m})>0$ is a constant.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Lemma 2.1
  • Definition 2.3
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 51 more