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Quantitative unique continuation on conic manifolds

Ruoyu P. T. Wang

Abstract

This expository note, written for the proceedings of ICCM 2023, presents recent work [arXiv:2004.13894]. We particularly prove an Carleman estimate on conic manifolds, using a multiple-weight Carleman argument.

Quantitative unique continuation on conic manifolds

Abstract

This expository note, written for the proceedings of ICCM 2023, presents recent work [arXiv:2004.13894]. We particularly prove an Carleman estimate on conic manifolds, using a multiple-weight Carleman argument.
Paper Structure (5 sections, 4 theorems, 42 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Local Carleman estimates
  3. Carleman estimates with multiple weights
  4. Weight functions construction
  5. Application: logarithmic decay

Key Result

Proposition 1.1

Let $M$ be compact and smooth, and $\Omega\subset M$ be an open set. Then any eigenfunction of $\Delta$ satisfies the unique continuation property: Moreover, we have the quantitative estimate, called the Carleman estimate, uniformly for $u\in H^2(M)$, where $C=C(M, \Omega)$ does not depend on $h$.

Figures (1)

  • Figure 1: Cutting up conic ends into sectors, where the dots stand for the critical points of weight functions.

Theorems & Definitions (5)

  • Proposition 1.1: Carleman estimates
  • Theorem 1: Carleman estimates on conic manifolds, wan20
  • Definition 3.1: Compatibility
  • Theorem 2: Global Carleman estimates with multiple weights, wan20
  • Corollary 5.1: Logarithmic decay, wan20