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Entanglement principle for fractional Laplacian on hyperbolic spaces and applications to inverse problem

Yi-Hsuan Lin

Abstract

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space $\mathbb H^n$, $n\ge 2$. More precisely, we prove that if finitely many distinct noninteger powers of $-Δ_{\mathbb H^n}$, acting on functions that vanish on a common nonempty open set, satisfy a linear dependence relation on that set, then each of these functions must vanish identically on $\mathbb H^n$. This extends the recently developed entanglement principle for the fractional Laplacian on $\mathbb R^n$ to the negatively curved setting of hyperbolic space. As an application, we derive global uniqueness results for inverse problems associated with fractional polyharmonic equations on $\mathbb H^n$, including a fractional Calderón problem. The proof relies on the heat semigroup representation of fractional powers together with sharp global heat kernel estimates on hyperbolic space.

Entanglement principle for fractional Laplacian on hyperbolic spaces and applications to inverse problem

Abstract

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space , . More precisely, we prove that if finitely many distinct noninteger powers of , acting on functions that vanish on a common nonempty open set, satisfy a linear dependence relation on that set, then each of these functions must vanish identically on . This extends the recently developed entanglement principle for the fractional Laplacian on to the negatively curved setting of hyperbolic space. As an application, we derive global uniqueness results for inverse problems associated with fractional polyharmonic equations on , including a fractional Calderón problem. The proof relies on the heat semigroup representation of fractional powers together with sharp global heat kernel estimates on hyperbolic space.
Paper Structure (14 sections, 12 theorems, 180 equations)

This paper contains 14 sections, 12 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Mathematical models
  3. Entanglement principle for the fractional Laplace operator
  4. An application: the fractional Calderón problem
  5. Preliminaries
  6. The fractional Laplacian on hyperbolic spaces
  7. The Helgason--Fourier transform
  8. The singular integral
  9. The heat semigroup
  10. Fractional Sobolev spaces
  11. The well-posedness
  12. The DN map
  13. Entanglement principle
  14. Application to inverse problems

Key Result

Theorem 1.1

Let $O\subset {\mathbb H}^n$ be a nonempty open set with $n\ge 2$. Let $N\in {\mathbb N}$ and $\{s_k\}_{k=1}^N\subset (0,\infty)\setminus {\mathbb N}$ satisfy Assumption $(H)$. Assume that $\{u_k\}_{k=1}^N \subset H^{-r}({\mathbb H}^n)$ for some $r\ge 0$, and let for some $\gamma>\frac{n-1}{2}$. Suppose that for each $k=1,\dots,N$, If for some $\{b_k\}_{k=1}^N\subset {\mathbb C}\setminus\{0\}$,

Theorems & Definitions (25)

  • Theorem 1.1: Entanglement principle
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1: BGS_15
  • Lemma 2.2
  • proof
  • Lemma 2.3: Well-posedness of the exterior Dirichlet problem
  • proof
  • Lemma 2.4: DN map on $\mathbb H^n$
  • proof
  • ...and 15 more