Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability

Technical Summary: Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability

Problem Statement
This study addresses the mathematical analysis of high-dimensional degenerate parabolic equations of the form:
$$\partial_t y - \text{div}(A\nabla y) = f \quad \text{in } Q = \Omega \times (0, T),$$
subject to Dirichlet boundary conditions $y=0$ on $\partial Q$ and initial data $y(0)=y_0$. The domain $\Omega \subseteq \mathbb{R}^{N-1} \times (0, +\infty)$ is a convex domain, and the diffusion matrix $A$ is diagonal, defined as $A = \text{diag}(1, \dots, 1, x_N^\alpha)$ with $\alpha \in (0, 1)$. Consequently, the equation degenerates on the boundary segment $\Gamma_0^N = \partial \Omega \cap (\mathbb{R}^{N-1} \times \{0\})$.

The primary objective is to establish the well-posedness of this equation and, more critically, to derive a boundary observability inequality. A significant challenge in higher-dimensional degenerate problems is the lack of appropriate integration-by-parts formulas when degeneracy occurs only on a portion of the boundary (rather than the entire boundary). Traditional methods, such as the method of separation of variables or specific structural assumptions on the coefficient matrix (e.g., degeneracy at the first eigenvalue across the whole boundary), are insufficient for the general case examined here.

Methodology: Shape Design Approach
The authors employ a shape design methodology to circumvent the difficulties associated with the degenerate boundary. The core strategy involves approximating the degenerate parabolic equation on the domain $\Omega$ by a sequence of uniformly parabolic equations defined on a family of subdomains $\{\Omega_\delta\}_{\delta > 0}$.

1. Domain Approximation: The authors construct a sequence of convex subdomains $\Omega_\delta$ such that $\Omega_\delta \subset \Omega$ and $\Omega_\delta$ stays away from the degenerate boundary segment $\Gamma_0^N$ by a distance proportional to $\delta$. Specifically, $\Omega_\delta$ is defined to exclude the region where $x_N < \delta$.
2. Approximating System: On each $\Omega_\delta$, the equation becomes uniformly parabolic because the coefficient $x_N^\alpha$ is bounded away from zero. This allows for the application of standard regularity theory and integration-by-parts techniques that are otherwise invalid on the original degenerate domain.
3. Extension and Convergence: Solutions $y_\delta$ on $\Omega_\delta$ are extended by zero to the entire domain $\Omega$ (denoted as $E y_\delta$). The authors prove that as $\delta \to 0$, these extended solutions converge to the weak solution of the original degenerate equation in appropriate weighted Sobolev spaces.
4. Carleman Estimates: For the approximating uniformly parabolic equations, the authors derive Carleman estimates with weights that depend on the parameter $\delta$. A crucial technical requirement is ensuring that the constants in these estimates remain uniform with respect to $\delta$. This uniformity is essential for passing to the limit and obtaining estimates for the degenerate case.

Key Contributions and Results

• Well-Posedness: The paper establishes the existence and uniqueness of weak solutions for the degenerate parabolic equation in the space $L^2(0, T; H^1_0(\Omega; w))$ with time derivatives in $L^2(0, T; H^{-1}(\Omega; w))$. It further provides higher regularity results for solutions with smoother initial data.
• Convergence of Approximations (Theorem 1.1): The authors rigorously demonstrate that the sequence of solutions to the uniformly parabolic approximations converges strongly in $L^2(Q)$ and weakly in weighted Sobolev spaces to the solution of the degenerate equation. Crucially, they prove the strong convergence of the normal derivatives $\frac{\partial y_\delta}{\partial \nu}$ to $\frac{\partial y}{\partial \nu}$ on the non-degenerate part of the boundary $\Gamma_+ = \{x \in \partial \Omega : A(x)\nu(x) \cdot e_N > 0\}$.
• Boundary Observability Inequality (Theorem 1.2 / Theorem 5.1): The main application of the shape design framework is the derivation of a boundary observability inequality for the degenerate equation:
  $$\int_\Omega y^2(0) dx \leq C \int_{\Gamma_+ \times (0, T)} \left| \frac{\partial y}{\partial \nu} \right|^2 dS dt.$$
  This inequality holds with a constant $C$ that depends only on $\alpha$, $T$, and the geometry of the domain, but is independent of the specific degeneracy parameter in a way that allows the limit process.
• Uniformity of Constants: A significant technical contribution is the proof that the constants in the Carleman estimates and the subsequent observability inequalities are uniform with respect to the approximation parameter $\delta$. This distinguishes the approach from methods that might yield constants blowing up as the domain approaches the degenerate configuration.

Significance and Claims
The paper claims that the shape design method offers a distinct advantage over existing techniques like cut-off methods or separation of variables. While cut-off techniques typically address issues within a single evolving system, the shape design approach utilizes a collection of distinct systems (the approximating domains) to bypass the need for integration by parts on the degenerate boundary.

The authors emphasize that their method does not rely on specific geometric assumptions about the domain (beyond convexity and smoothness) or structural assumptions on the coefficient matrix that force degeneracy across the entire boundary. This makes the approach applicable to a broader class of high-dimensional degenerate parabolic equations where degeneracy is localized to a segment of the boundary. The results provide a theoretical foundation for boundary controllability and observability in these challenging settings, where traditional integration-by-parts arguments fail.