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The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds

Yi-Hsuan Lin

Abstract

We consider the fractional anisotropic Calderón problem for the nonlocal parabolic equation $(\partial_t -Δ_g)^s u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calderón problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.

The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds

Abstract

We consider the fractional anisotropic Calderón problem for the nonlocal parabolic equation () on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calderón problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.

Paper Structure

This paper contains 6 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The nonlocal parabolic operator on closed Riemannian manifolds
  3. The well-posedness
  4. The Calderón problem
  5. Auxiliary lemmas
  6. Proof of Theorem \ref{['thm: main']}

Key Result

Theorem 1.1

Given $s\in (0,1)$, let $(M_1,g_1)$ and $(M_2,g_2)$ be closed connected smooth Riemannian manifolds of $\dim M_1= \dim M_2\geq 2$. Let $\mathcal{O}_j \subset M_j$ be a nonempty open sets (for $j=1,2$) such that Suppose that where $S_{M_j,g_j,\mathcal{O}_j}$ denotes the local source-to-solution map of for $j=1,2$. Then there exists a diffeomorphism $\Phi: M_1\to M_2$ such that $\Phi^\ast g_2=g_1

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: Representation formula
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1: Well-posedness
  • proof
  • ...and 7 more