Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary Control and Calderón type Inverse Problems in Non-local heat equation

Saumyajit Das

TL;DR

This work analyzes two non-local parabolic models driven by the fractional Laplacian $(-\Delta)^a$ with $a\in(1/2,1)$ and smooth potentials, focusing on density of solution spaces on an interior time-slice and on the boundary, and on Calderón-type inverse problems for recovering the potential $q$. The authors leverage the Pohozaev identity, a framework of $\mu$-transmission spaces, and dual/variational density methods to obtain both qualitative and quantitative density results, enabling reconstruction of $q$ from interior or boundary data and establishing equivalence between the two inverse problems. A key contribution is the boundary-non-vanishing property of eigenfunctions, derived via Pohozaev-type identities for an elliptic problem, which holds without restricting $a$. Collectively, the results advance nonlocal parabolic inverse problems, boundary controllability, and the utility of boundary traces in transmission-space analyses.

Abstract

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary data. In both the cases, we assume the non-local exponent $a\in(\frac{1}{2},1)$. We explore both the qualitative and quantitative aspects of the approximations. Additionally, we address Calderón-type inverse problems for these parabolic models, where we recover the potentials by analyzing the solutions either on the boundary or at a particular time slice. In both the density results and the Calderón type inverse problems, the Pohozaev identity plays a crucial role. Finally, in the last section, we apply the Pohozaev identity to a specific elliptic eigenvalue problem and demonstrate that the eigenfunctions, when divided by an appropriate power of the distance function, can not vanish on any non-empty open subset of the boundary. This particular eigenvalue problem does not need any restriction on the non-local exponent.

Boundary Control and Calderón type Inverse Problems in Non-local heat equation

TL;DR

This work analyzes two non-local parabolic models driven by the fractional Laplacian with and smooth potentials, focusing on density of solution spaces on an interior time-slice and on the boundary, and on Calderón-type inverse problems for recovering the potential . The authors leverage the Pohozaev identity, a framework of -transmission spaces, and dual/variational density methods to obtain both qualitative and quantitative density results, enabling reconstruction of from interior or boundary data and establishing equivalence between the two inverse problems. A key contribution is the boundary-non-vanishing property of eigenfunctions, derived via Pohozaev-type identities for an elliptic problem, which holds without restricting . Collectively, the results advance nonlocal parabolic inverse problems, boundary controllability, and the utility of boundary traces in transmission-space analyses.

Abstract

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary data. In both the cases, we assume the non-local exponent . We explore both the qualitative and quantitative aspects of the approximations. Additionally, we address Calderón-type inverse problems for these parabolic models, where we recover the potentials by analyzing the solutions either on the boundary or at a particular time slice. In both the density results and the Calderón type inverse problems, the Pohozaev identity plays a crucial role. Finally, in the last section, we apply the Pohozaev identity to a specific elliptic eigenvalue problem and demonstrate that the eigenfunctions, when divided by an appropriate power of the distance function, can not vanish on any non-empty open subset of the boundary. This particular eigenvalue problem does not need any restriction on the non-local exponent.
Paper Structure (9 sections, 15 theorems, 169 equations)

This paper contains 9 sections, 15 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Fractional Laplacian and fractional Sobolev space
  3. Wellposedness of the solution
  4. $\mu$ transmission spaces
  5. Boundary values of $\mu$ transmission spaces, Dirichlet trace
  6. Density Results:
  7. Calderón type Inverse Problems:
  8. Further result:
  9. Appendix

Key Result

Theorem 1

Let $a\in(0,1)$ and $B_1$ is the unit ball in $\mathbb{R}^N$ centered at origin. Let $W\subset \mathbb{R}^N$ be a non-empty open set such that $\overline{B}_1\cap \overline{W}$ is empty. Let for $f\in C_c^{\infty}((-1,1)\times W)$, the function $u_f:(-1,1)\times\mathbb{R}^N\to \mathbb{R}$ be the sol Then the solution set is dense in $\mathrm{L}^2((-1,1)\times B_1)$.

Theorems & Definitions (23)

  • Theorem 1: Ruland2017QuantitativeAP
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: fernandez2016boundary
  • Lemma 1
  • Lemma 2: claus2020realization, grubb2018regularity
  • Theorem 7
  • Theorem 8: grubb2023resolvents
  • ...and 13 more