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An inverse problem for a nonlinear biharmonic operator

Janne Nurminen, Suman Kumar Sahoo

TL;DR

This work develops an inverse problem framework for a general nonlinear biharmonic operator $\mathcal{L}_{Q}u=\Delta^2u+Q(x,u,\nabla u,\Delta u)$ under Navier boundary data. The authors implement a two-pronged strategy: first, construct a smooth solution map around a fixed solution via a contraction-based fixed-point argument; second, apply first-order linearization together with Runge approximation to compare two nonlinearities from local Cauchy data. They prove that if two nonlinearities share a common solution and yield locally inclusive Cauchy data, then they agree near that solution up to a gauge transformation; when boundary data coincide in the Navier map, the nonlinear terms are determined in a neighborhood. The results extend Isakov-type approaches to fourth-order equations, broadening the class of nonlinearities recoverable from local boundary measurements and highlighting the role of Runge approximation in higher-order inverse problems.

Abstract

An inverse problem for a nonlinear biharmonic operator is under consideration in the spirit of Isakov (1993) and Johansson-Nurminen-Salo (2023). We prove that a general nonlinear term of the $Q= Q(x,u, \nabla u, Δu)$ associated to a nonlinear biharmonic operator can be recovered from the local Cauchy data set. The proof uses first order linearization method, Runge approximation, and uniqueness results for the linearized inverse problem.

An inverse problem for a nonlinear biharmonic operator

TL;DR

This work develops an inverse problem framework for a general nonlinear biharmonic operator under Navier boundary data. The authors implement a two-pronged strategy: first, construct a smooth solution map around a fixed solution via a contraction-based fixed-point argument; second, apply first-order linearization together with Runge approximation to compare two nonlinearities from local Cauchy data. They prove that if two nonlinearities share a common solution and yield locally inclusive Cauchy data, then they agree near that solution up to a gauge transformation; when boundary data coincide in the Navier map, the nonlinear terms are determined in a neighborhood. The results extend Isakov-type approaches to fourth-order equations, broadening the class of nonlinearities recoverable from local boundary measurements and highlighting the role of Runge approximation in higher-order inverse problems.

Abstract

An inverse problem for a nonlinear biharmonic operator is under consideration in the spirit of Isakov (1993) and Johansson-Nurminen-Salo (2023). We prove that a general nonlinear term of the associated to a nonlinear biharmonic operator can be recovered from the local Cauchy data set. The proof uses first order linearization method, Runge approximation, and uniqueness results for the linearized inverse problem.

Paper Structure

This paper contains 9 sections, 21 theorems, 138 equations.

Table of Contents

  1. introduction and main results
  2. State of the Art
  3. A smooth solution map
  4. Estimates for solutions based on their Cauchy data
  5. Controlling Cauchy data
  6. First linearization
  7. proof of main results
  8. Runge approximation
  9. computational details

Key Result

Theorem 1.1

Let $n \geq 2$ and $w$ solve $\mathcal{L}_{Q_1} w = \mathcal{L}_{Q_2} w = 0$ in $\Omega$, where $Q_1$ and $Q_2$ satisfy eq_reg_Q. Suppose for some $\delta, C > 0$ we have $C_{Q_1}^{w, \delta} \subseteq C_{Q_2}^{0, C}.$ Then there exists $\epsilon > 0$ such that for $x \in \overline{\Omega}$, $|\lambda| \leq \epsilon$, and $\overline{\lambda} = (\lambda, \ldots, \lambda) \in \mathbb{R}^n$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 29 more