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Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

R. M. Brown, L. D. Gauthier

TL;DR

This work extends Haberman–Tataru's averaging technique to polyharmonic operators to achieve uniqueness in inverse boundary value problems with very rough first-order perturbations. By constructing CGO solutions in $X^\lambda$ spaces and employing an averaged analysis, the authors prove that the lower-order coefficients $Q$ and $q$ are uniquely determined by boundary data, even when $Q\in\tilde{W}^{-s+1,p}$ and $q\in\tilde{W}^{-s,p}$ with $s< m/2+1$. The main result shows that equality of Cauchy data or the associated bilinear forms implies $Q^1=Q^2$ and $q^1=q^2$ for $m\ge2$ and $d\ge3$. This advances the theory of inverse problems for polyharmonic operators by relaxing regularity assumptions, bringing results closer to optimal Sobolev thresholds and highlighting the central role of averaging in high-order contexts.

Abstract

We consider inverse boundary value problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

TL;DR

This work extends Haberman–Tataru's averaging technique to polyharmonic operators to achieve uniqueness in inverse boundary value problems with very rough first-order perturbations. By constructing CGO solutions in spaces and employing an averaged analysis, the authors prove that the lower-order coefficients and are uniquely determined by boundary data, even when and with . The main result shows that equality of Cauchy data or the associated bilinear forms implies and for and . This advances the theory of inverse problems for polyharmonic operators by relaxing regularity assumptions, bringing results closer to optimal Sobolev thresholds and highlighting the central role of averaging in high-order contexts.

Abstract

We consider inverse boundary value problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

Paper Structure

This paper contains 7 sections, 21 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. CGO solutions and $X^ \lambda$ spaces
  4. Average Estimate
  5. Cauchy data, bilinear forms and Dirichlet to Neumann maps
  6. Main Result
  7. Sobolev spaces

Key Result

Theorem 1.3

Fix $m \geq 2$ and $d \geq 3$ and let $\Omega \subset \mathbb{R} ^d$ be a bounded domain with smooth boundary. Let $L_j = ( -\Delta)^m + Q^j\cdot D + q^j$ with $q^j \in \tilde{W}^ { -s,p} ( \Omega)$ and each component $Q^j$ lies in $\tilde{W}^{ -s+1,p}(\Omega)$ with $s < m/2 +1$, $1/p +(s-m)/d <0$,

Theorems & Definitions (41)

  • Theorem 1.3
  • Lemma 3.5: MR3024091
  • Lemma 3.6
  • proof
  • Lemma 3.9
  • proof
  • Theorem 3.11
  • proof
  • Corollary 3.13
  • proof
  • ...and 31 more