Inverse problems for semilinear elliptic equations with low regularity

Technical Summary: Inverse Problems for Semilinear Elliptic Equations with Low Regularity

Problem Statement
This article addresses inverse boundary value problems for semilinear elliptic equations of the form:
$$\Delta u(x) + a(x, u(x)) = 0 \quad \text{in } \Omega,$$
where $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain with a smooth boundary, and $a(x, t)$ represents a nonlinearity. The objective is to determine the nonlinearity $a(x, t)$ from boundary measurements. Specifically, the authors investigate whether $a(x, t)$ can be uniquely recovered in a neighborhood of a fixed solution $w$ using the Cauchy data set of solutions close to $w$.

A primary challenge addressed is the regularity of the nonlinearity. Previous results, such as those in [JNS23], required $a(x, t)$ to be $C^{1,\alpha}$ in $x$ and $C^3$ in $t$. This paper aims to lower these regularity assumptions to $L^r$ in $x$ (with $r > n/2$) and $C^{1,\alpha}$ in $t$, while simultaneously simplifying the existing proofs. The study does not assume the well-posedness of the Dirichlet problem a priori; instead, it formulates the problem using Cauchy data sets to accommodate potential obstructions.

Methodology
The approach relies on a combination of first-order linearization, implicit function theorems in Banach spaces, and Runge approximation arguments. The methodology is structured as follows:

1. Linearization and Solvability: The equation is linearized around a fixed solution $w$, yielding $\Delta v + qv = 0$ where $q(x) = \partial_u a(x, w(x))$. Since $q \in L^r(\Omega)$ with $r > n/2$, the standard Dirichlet problem may not be well-posed due to a finite-dimensional kernel $N_q$. The authors construct a subspace $D_q$ of the boundary trace space to modify the boundary data, ensuring solvability for the linearized equation (Proposition 2.1). This replaces the need for the normal derivative of eigenfunctions used in higher-regularity settings.

2. Solution Maps:

   • Map $S$: A $C^1$ map $S(v)$ is constructed that parametrizes solutions $u$ of the nonlinear equation near $w$ in terms of solutions $v$ of the linearized equation. Unlike previous works that used a Banach fixed-point argument followed by an implicit function theorem, this paper establishes both existence and $C^1$ smoothness of $S$ via a single application of the implicit function theorem (Proposition 3.1).
   • Map $T$: To compare two different nonlinearities $a_1$ and $a_2$, a second map $T_{a_2, w_1}$ is defined. This map takes a linear solution $v$ associated with $a_1$ and produces a solution $u_2$ for $a_2$ that shares the same Cauchy data as the solution generated by $S_{a_1, w_1}(v)$. Proving that $T$ is $C^1$ requires a unique continuation argument and a specific projection operator derived from a fourth-order equation (Proposition 4.1, Lemma 4.5).

3. Integral Identity and Uniqueness: By assuming the inclusion of Cauchy data sets ($C_{w, \delta}^{a_1} \subseteq C_{0, C}^{a_2}$), the authors derive an integral identity involving the difference of the linearized potentials $\partial_u a_1$ and $\partial_u a_2$. Using the completeness of products of solutions to linear Schrödinger equations (results from [Cha90; Nac92; DKS13; BTW20]) and unique continuation principles, they show that the derivatives of the nonlinearities must coincide near the solution.

4. Runge Approximation: To extend the local determination of the nonlinearity to a full neighborhood of the solution values, a Runge approximation argument is employed (Section 6). This demonstrates that solutions to the linearized equation with $L^r$ potentials can approximate any value in a neighborhood of the solution point, allowing the recovery of $a(x, t)$ for a range of $t$.

Key Contributions and Results

• Reduced Regularity Assumptions: The main theorems (Theorems 1.1 and 1.2) establish uniqueness results for nonlinearities $a \in L^r(\Omega, C^{1,\alpha}(\mathbb{R}))$ with $r > n/2$. This improves upon the $C^{1,\alpha}$ in $x$ and $C^3$ in $t$ requirements of [JNS23].
• Simplified Proofs: The paper provides a streamlined proof for the existence and smoothness of the solution map $S$ by utilizing the implicit function theorem directly, avoiding the more complex fixed-point arguments used previously.
• Handling Low Regularity in Linearization: The authors introduce a subspace $D_q$ to handle the solvability of the linearized equation when the potential $q$ is only in $L^r$, overcoming the issue that the normal derivatives of eigenfunctions may not lie in the natural trace space for low-regularity potentials.
• Main Theorems:
  • Theorem 1.1: If two nonlinearities $a_1, a_2$ share a common solution $w$ and have the same Cauchy data for solutions near $w$, then $a_1(x, t) = a_2(x, t)$ for $t$ near $w(x)$.
  • Theorem 1.2: If $a_1$ and $a_2$ do not necessarily share a solution, but their Cauchy data sets near a solution $w_1$ of $a_1$ are related, then $a_1$ is determined up to a gauge transformation $T_\phi a_2(x, t) = \Delta \phi(x) + a_2(x, t + \phi(x))$, where $\phi$ vanishes on the boundary with its normal derivative.
• Linear Case Recovery: The results cover the linear case $a(x, u) = q(x)u$, recovering potentials in $L^r(\Omega)$ for $r > n/2$ (and $r \geq 2$ for $n=2,3$). The authors note this misses the endpoint $r=n/2$ for $n \geq 3$ and specific $L^{4/3+\epsilon}$ results for $n=2$ found in other literature.

Significance and Claims
The paper claims to provide low-regularity counterparts to recent results in [JNS23], demonstrating that the nonlinearity can be uniquely determined (up to gauge) from boundary measurements without requiring high smoothness in the spatial variable $x$. The authors emphasize that their method relies on first-order linearization and Runge approximation, avoiding the higher-order linearization techniques that typically only recover Taylor series at $t=0$.

The work highlights that the reachable set for these inverse problems is always an open set, implying that the nonlinearity can be determined in a neighborhood of the solution. The authors explicitly state that they do not claim to determine $a(x, t)$ globally on $\Omega \times \mathbb{R}$, as this is generally impossible without further assumptions, but rather in the neighborhood of the specific solution trajectory. The results are presented as improvements in regularity and simplifications in proof structure for existing inverse problem frameworks.