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Inverse problems for semilinear elliptic equations with low regularity

David Johansson, Janne Nurminen, Mikko Salo

TL;DR

This work advances inverse boundary value problems for semilinear elliptic equations with low spatial regularity by proving that a general nonlinearity $a(x,u)$ is uniquely determined near a fixed solution from boundary measurements, up to a gauge when no common solution is available. The approach combines first linearization with Runge approximation in a framework where the nonlinearity satisfies $a\in L^r(\Omega, C^{1,\alpha}(\mathbb{R}))$ with $r>n/2$, replacing the prior $C^{1,\alpha}$-in-x assumptions. The paper constructs a linearized solvability theory with a finite-dimensional obstruction $N_q$, introduces $D_q$ as a boundary data correction space, and builds $C^1$-regular nonlinear solution maps $S_{a,w}$ and $T_{a,w}$ to compare nonlinearities via boundary data inclusions. Using integral identities together with the completeness of products of linear solutions and Runge approximation, the authors deduce local equality of nonlinearities or a gauge-equivalence relation, thereby extending Johansson–Nurminen–Salo-type results to lower regularity. The Runge approximation result for $L^r$ potentials is a key technical tool enabling the realization of interior values and completing the inverse analysis in arbitrary subregions.

Abstract

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in Johansson-Nurminen-Salo (2023).

Inverse problems for semilinear elliptic equations with low regularity

TL;DR

This work advances inverse boundary value problems for semilinear elliptic equations with low spatial regularity by proving that a general nonlinearity is uniquely determined near a fixed solution from boundary measurements, up to a gauge when no common solution is available. The approach combines first linearization with Runge approximation in a framework where the nonlinearity satisfies with , replacing the prior -in-x assumptions. The paper constructs a linearized solvability theory with a finite-dimensional obstruction , introduces as a boundary data correction space, and builds -regular nonlinear solution maps and to compare nonlinearities via boundary data inclusions. Using integral identities together with the completeness of products of linear solutions and Runge approximation, the authors deduce local equality of nonlinearities or a gauge-equivalence relation, thereby extending Johansson–Nurminen–Salo-type results to lower regularity. The Runge approximation result for potentials is a key technical tool enabling the realization of interior values and completing the inverse analysis in arbitrary subregions.

Abstract

We show that a general nonlinearity is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in Johansson-Nurminen-Salo (2023).
Paper Structure (6 sections, 18 theorems, 127 equations)

This paper contains 6 sections, 18 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Solvability for linear equations
  3. Solution map for nonlinear equation
  4. Second solution map
  5. Inverse problem
  6. Runge approximation

Key Result

Theorem 1.1

Let $a_1, a_2 \in L^r(\Omega, C^{1,\alpha}(\mathbb{R}))$ where $r > n/2$, $r \geq 2$ and $\alpha > 0$. Suppose that $w \in W^{2,r}(\Omega)$ solves $\Delta w + a_j(x,w) = 0$ in $\Omega$ for $j = 1,2$. If for some $\delta, C > 0$ one has then there is $\varepsilon > 0$ such that

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['prop_linearized_ri_wonep']}
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 27 more