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Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator

Russell M. Brown, Landon Gauthier, Daniel Faraco

TL;DR

The paper tackles the inverse boundary value problem for a higher-order elliptic operator with principal part $(-\Delta)^m$ and lower-order symmetric tensor coefficients, showing that the coefficients are uniquely determined by boundary data under low regularity. The authors construct CGO solutions for the conjugated operator, derive a main bilinear-form equation, and apply a novel tensor-analytic framework to reduce the problem to a Fourier-analytic set of constraints. A tensor Structure Theorem and a sequence of directional-derivative analyses yield that all coefficient tensors must vanish when the bilinear form agrees, proving injectivity of the coefficient-to-boundary map in two regimes; they also outline open questions on optimal regularity and extensions. These results advance the Calderón-type inverse problem for polyharmonic operators and refine the understanding of how boundary measurements determine complex, high-order perturbations.

Abstract

We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator $(-Δ)^m$ with $ m \geq 2$. We show that the map from the coefficients to a certain bilinear form is injective. We have a particular focus on obtaining these results under lower regularity on the coefficients. It is known that knowledge of this bilinear form is equivalent to a knowledge of a Dirichlet to Neumann map or the Cauchy data for solutions.

Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator

TL;DR

The paper tackles the inverse boundary value problem for a higher-order elliptic operator with principal part and lower-order symmetric tensor coefficients, showing that the coefficients are uniquely determined by boundary data under low regularity. The authors construct CGO solutions for the conjugated operator, derive a main bilinear-form equation, and apply a novel tensor-analytic framework to reduce the problem to a Fourier-analytic set of constraints. A tensor Structure Theorem and a sequence of directional-derivative analyses yield that all coefficient tensors must vanish when the bilinear form agrees, proving injectivity of the coefficient-to-boundary map in two regimes; they also outline open questions on optimal regularity and extensions. These results advance the Calderón-type inverse problem for polyharmonic operators and refine the understanding of how boundary measurements determine complex, high-order perturbations.

Abstract

We consider an inverse problem for a higher order elliptic operator where the principal part is the polyharmonic operator with . We show that the map from the coefficients to a certain bilinear form is injective. We have a particular focus on obtaining these results under lower regularity on the coefficients. It is known that knowledge of this bilinear form is equivalent to a knowledge of a Dirichlet to Neumann map or the Cauchy data for solutions.
Paper Structure (12 sections, 20 theorems, 147 equations)

This paper contains 12 sections, 20 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Constructing CGO solutions
  4. From the Bilinear Form to equations on the coefficients
  5. Uniqueness from the main equation
  6. Tensor Structure theorem
  7. Tensors vanishing on ${\cal{V}}$
  8. Proof of the structure theorem
  9. Conditions on directional derivatives on the Fourier side
  10. Proof of Theorem \ref{['thm:uniquenessmaster']}
  11. Further questions
  12. Some Tensor Algebra

Key Result

Theorem 1.2

Let $L_1$ and $L_2$ be as in e:IntroOp with coefficients $A^{(k)}_ \ell =( A^{(k)} _ {\ell,\alpha} )_ { |\alpha| =k}$, $\ell = 1,2$, $k=0,\dots, k_0$ and that the coefficients $A^{(k)}_\ell= 0$ for $k > k_0$. We give two cases for parameters that we will use in our hypotheses on the coefficients: We assume that the coefficients $A_\ell^{(k)} \in \tilde{W} ^ { k-s, p} (\Omega)$ for $j=0,\dots, k_

Theorems & Definitions (40)

  • Theorem 1.2
  • Proposition 2.14
  • proof
  • Proposition 2.21
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 30 more