Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data

1. Problem Statement

The paper investigates inverse problems associated with the spectral fractional Laplacian operator, denoted as $(-\Delta_D)^s$, defined on a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) with a smooth boundary. The study focuses on the inhomogeneous Dirichlet boundary value problem for the fractional Schrödinger equation:
$$((-\Delta_D)^s - q) u = 0 \quad \text{in } \Omega, \quad u|_{\partial \Omega} = g,$$
where $q \in L^\infty(\Omega)$ is an unknown potential and $g$ is the boundary data.

The primary objective is to determine the potential $q$ (or a set of coefficients in a higher-order problem) uniquely from boundary measurements. Specifically, the authors define a Dirichlet-to-Neumann (DN) map $\Lambda_q$ associated with this problem and analyze its linearization (the Fréchet derivative) at zero potential, denoted as $d\Lambda$. The core question is whether the map $q \mapsto \Lambda_q$ is injective and stable.

2. Methodology

The authors employ a combination of functional analysis, spectral theory, and the method of Complex Geometric Optics (CGO) solutions.

• Spectral Definition: Unlike the integral definition of the fractional Laplacian (often used in exterior data problems), the authors utilize the spectral definition based on the eigenvalues $\{\lambda_k\}$ and eigenfunctions $\{\phi_k\}$ of the classical Dirichlet Laplacian $-\Delta$. The operator is defined via the spectral decomposition:
  $$(-\Delta_D)^s v = \sum_{k=1}^\infty \lambda_k^s \langle v, \phi_k \rangle \phi_k.$$
  They rigorously define the operator on appropriate fractional Sobolev spaces ($H^s(\Omega)$, $H^{-s}(\Omega)$) and handle the inhomogeneous boundary conditions using a Poisson operator approach.

• Linearization and Born Approximation: The study focuses on the linearized inverse problem near $q=0$. The authors establish that the Fréchet derivative $d\Lambda$ corresponds to the Born approximation. They derive an Alessandrini identity linking the DN map to an integral involving the potential and solutions to the unperturbed equation:
  $$\langle h, (d\Lambda[q])g \rangle_{\partial \Omega} = -\int_\Omega q u_h u_g \, dx,$$
  where $u_g$ and $u_h$ are solutions to the homogeneous fractional equation with boundary data $g$ and $h$.

• Complex Geometric Optics (CGO): To prove uniqueness, the authors construct special solutions (CGO solutions) of the form $e^{\alpha \cdot x}$ (for the linearized problem) or more refined asymptotic expansions involving transport operators. These solutions are used to probe the integral identity, effectively recovering the Fourier transform of the potential.

• Density and Gauge Invariance: For the second main result, the authors address a higher-order inverse problem involving coefficients $\theta_2, \theta_1, \theta_0$ in a generalized equation. They utilize Runge approximation properties and density arguments to show that if the integral identity holds for all harmonic/biharmonic test functions, the coefficients must satisfy a specific "gauge" structure.

3. Key Contributions

A. Definition of the DN Map for Inhomogeneous Data

The paper provides a rigorous framework for the DN map in the context of the spectral fractional Laplacian with inhomogeneous Dirichlet data. This fills a gap in the literature, which has largely focused on exterior data problems (Fourier fractional Laplacian) or homogeneous boundary conditions.

B. Uniqueness and Stability for the Linearized Problem (Theorem 1.1)

The authors prove that the linearized DN map uniquely determines the potential $q$.

• Uniqueness: If $d\Lambda[q^{(1)}] = d\Lambda[q^{(2)}]$, then $q^{(1)} \equiv q^{(2)}$.
• Stability: They establish a logarithmic stability estimate (conditional stability):
  $$\|q^{(1)} - q^{(2)}\|_{H^{-1}(\mathbb{R}^n)} \le C w(\|d\Lambda[q^{(1)}] - d\Lambda[q^{(2)}]\|_*),$$
  where $w(t) \sim |\log t|^{-1}$. This is the standard stability rate for ill-posed inverse problems of this type.

C. Recovery of Coefficients up to Gauge (Theorem 1.4)

The paper addresses the recovery of coefficients in a generalized operator form:
$$\theta_2 : \nabla^{\otimes 2} u + \theta_1 \cdot \nabla u + \theta_0 u.$$
They prove that if the integral identity holds for all $u$ satisfying $\Delta^2 u = 0$ and $v$ satisfying $(-\Delta_D)^s v = 0$, then the coefficients are determined up to a natural gauge transformation generated by a function $w$:
$$\theta_2 = w I_n, \quad \theta_1 = 2\nabla w, \quad \theta_0 = \Delta w.$$

• Dimensional Distinction:
  • For $n \ge 3$, the gauge is identified exactly as above.
  • For $n = 2$, under the additional assumptions $\theta_0 = 0$ and $\text{tr}(\theta_2) = 0$, the authors conclude that $\theta_1 = 0$ and $\theta_2 = 0$ (uniqueness without gauge).

4. Key Results Summary

1. Theorem 1.1: Establishes the injectivity and logarithmic stability of the linearized DN map for the spectral fractional Schrödinger equation with inhomogeneous Dirichlet data. This extends previous results (which often used exterior data) to the interior boundary measurement setting.
2. Theorem 1.4: Solves an inverse problem for a second-order differential operator perturbed by the spectral fractional Laplacian. It characterizes the non-uniqueness (gauge invariance) of the coefficients $\theta_2, \theta_1, \theta_0$.
3. Density Result: The paper establishes an additional density result for the spectral fractional Laplacian, which is crucial for the Runge approximation arguments used in the proofs.

5. Significance and Impact

• Bridging a Gap: Most existing literature on fractional inverse problems relies on the exterior data formulation (where the fractional Laplacian is non-local and data is taken outside the domain). This paper successfully adapts the methodology to the spectral fractional Laplacian with interior (boundary) data, which is more relevant for certain physical applications where the domain is bounded and boundary measurements are the only available data.
• Rigorous Framework: The work provides precise definitions of the operators and DN maps in suitable Hilbert spaces, clarifying the relationship between the spectral fractional Laplacian and the classical Laplacian in the context of boundary value problems.
• Extension of Calderón's Problem: By solving the linearized version and establishing stability, the paper contributes to the modern extension of the classical Calderón problem to fractional operators, demonstrating that the non-local nature of the operator (via the Runge approximation) still allows for unique recovery even with standard boundary measurements.
• Gauge Analysis: The detailed analysis of the gauge invariance in Theorem 1.4 provides insight into the structural limitations of inverse problems involving higher-order terms coupled with fractional operators, distinguishing between dimensions $n=2$ and $n \ge 3$.

In conclusion, this paper advances the mathematical theory of inverse problems by rigorously treating the spectral fractional Laplacian with inhomogeneous boundary conditions, proving uniqueness and stability for the linearized problem, and characterizing the gauge freedom in related coefficient recovery problems.