Lipschitz stability for an inverse source problem of the wave equation with kinetic boundary conditions

Technical Summary: Lipschitz Stability for an Inverse Source Problem of the Wave Equation with Kinetic Boundary Conditions

Problem Statement
This paper addresses an inverse source problem for the wave equation defined on a bounded domain $\Omega \subset \mathbb{R}^n$ with a smooth boundary $\Gamma = \Gamma_0 \cup \Gamma_1$. The system incorporates mixed boundary conditions: a Dirichlet condition on $\Gamma_0$ and a dynamic (kinetic) boundary condition on $\Gamma_1$. The dynamic condition accounts for the momentum of the wave on the boundary and the bulk-surface interaction, modeled by the equation $\partial^2_t y_\Gamma - \delta \Delta_\Gamma y_\Gamma + d\partial_\nu y + q_\Gamma(x)y_\Gamma = g(x,t)$, where $y_\Gamma$ is the trace of the solution on the boundary, $\Delta_\Gamma$ is the Laplace-Beltrami operator, and $\partial_\nu$ denotes the normal derivative.

The primary objective is to establish global Lipschitz stability for the simultaneous determination of two unknown forcing terms, $f(x,t)$ in the bulk and $g(x,t)$ on the boundary $\Gamma_1$, from a single partial measurement of the boundary flux $\partial_\nu y$ on a subset $\gamma \subset \Gamma_0$. The source terms are assumed to belong to an admissible set $S(C_0)$ where their time derivatives are bounded by their initial values.

Methodology
The core methodology relies on the derivation of a sharp global Carleman estimate for the wave system with dynamic boundary conditions. The authors employ a refined approach involving the following steps:

1. Correction of Previous Weight Functions: The authors identify a critical flaw in the weight function $\zeta(x,t) = |x-x_0|^2 - \beta t^2 + C_0$ used in prior work by Gal and Tebou [16]. Specifically, the identity $\nabla^2_\Gamma \zeta = 2I$ claimed in [16] is shown to be incorrect for standard domains (e.g., a sphere). To resolve this, the authors introduce a new weight function $\psi(x,t) = \psi_0(x) - \beta t^2 + C_1$, where $\psi_0(x) = \mu^2(x)$ and $\mu$ is the Minkowski (gauge) function associated with a convex set $\Omega_1$ contained within $\Omega$. This construction ensures that $\nabla_\Gamma \psi = 0$ and $\nabla^2_\Gamma \psi = 0$ on $\Gamma_1$, satisfying the necessary pseudo-convexity conditions without relying on the erroneous identity.

2. Carleman Estimate Derivation: Using the new weight function, the authors derive a Carleman estimate (Theorem 3.1) for the system. A key technical achievement is the ability to keep the term involving the tangential gradient $|\nabla_\Gamma y_\Gamma|^2$ on the left-hand side of the inequality, provided the condition $\delta > d$ holds. This is achieved by carefully selecting the parameter $\beta$ and utilizing the properties of the new weight function, which avoids the need for cut-off functions in time that were previously required.

3. Stability Proof: The inverse problem is solved by extending a recent argument (avoiding cut-off functions) to the dynamic boundary setting. By differentiating the system with respect to time and applying the derived Carleman estimate to the time-differentiated solution, the authors relate the $L^2$ norms of the source terms to the observation of the time derivative of the boundary flux $\partial_t \partial_\nu y$.

4. Controllability Application: As a secondary application, the established Carleman estimate is used via a standard duality argument to prove exact boundary controllability for the system with a control acting only on the Dirichlet portion $\gamma \subset \Gamma_0$.

Key Results

• Theorem 3.1 (Carleman Estimate): The paper establishes a global Carleman estimate for the wave equation with dynamic boundary conditions. This estimate corrects and improves upon the result in [16] by using a valid weight function that ensures pseudo-convexity on the boundary. Crucially, under the condition $\delta > d$, the estimate absorbs the tangential gradient term $|\nabla_\Gamma y_\Gamma|^2$ into the left-hand side, providing a sharper inequality.
• Theorem 4.1 (Lipschitz Stability): For a time $T$ greater than a specific minimal time $T^*$ (dependent on the geometry and wave speeds), and assuming $\delta > d$, the authors prove a global Lipschitz stability estimate:
  $$\|f\|_{L^2} + \|g\|_{L^2} \leq C \|\partial_t \partial_\nu y\|_{L^2(\gamma \times (0,T))}$$
  This result guarantees the unique and stable recovery of the source terms from a single boundary measurement.
• Theorem 5.5 (Exact Controllability): The system is shown to be exactly controllable in time $T > 2T^*$ using a single control force acting on the Dirichlet boundary subset $\gamma$. This improves upon previous results that required two control forces.

Significance and Claims
The authors claim that this work provides the first global Lipschitz stability result for an inverse hyperbolic problem with general dynamic/Dirichlet boundary conditions. The significance lies in:

1. Rigorous Correction: The paper corrects a fundamental error in the weight function analysis of the seminal work by Gal and Tebou [16], which had relied on an incorrect Hessian identity for the boundary.
2. Sharpness: The derived Carleman estimate is "sharp" in the sense that it places the tangential derivative terms on the left-hand side (under the condition $\delta > d$), whereas previous estimates placed them on the right-hand side or required additional assumptions.
3. Methodological Advancement: The paper demonstrates that a different weight function, originally used for Schrödinger equations, can be successfully adapted to wave equations with dynamic boundary conditions, overcoming difficulties related to the coupling of bulk and surface dynamics.
4. Optimality: The controllability result is noted as sharp because it achieves exact controllability with a single control on the Dirichlet part, contrasting with earlier works that utilized two controls.

The paper concludes by identifying open problems, specifically the case where $\delta \leq d$ (where the current Carleman estimate fails to absorb certain terms) and the extension of these results to the case where dynamic boundary conditions are imposed on the entire boundary $\Gamma$.