Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

1. Problem Statement and Context

The paper investigates shape optimization problems involving two fundamental functionals defined on open, bounded, convex sets $\Omega \subset \mathbb{R}^n$ with Robin boundary conditions ($\frac{\partial u}{\partial \nu} + \beta u = 0$ on $\partial\Omega$, where $\beta > 0$):

1. Robin Torsion ($T_\beta(\Omega)$): The $L^1$-norm of the solution to the torsion problem $-\Delta u = 1$. It admits a variational characterization:
   $$T_\beta(\Omega) = \sup_{v \in H^1(\Omega)} \frac{\left(\int_\Omega v \, dx\right)^2}{\int_\Omega |\nabla v|^2 \, dx + \beta \int_{\partial\Omega} v^2 \, d\mathcal{H}^{n-1}}$$
2. First Robin Eigenvalue ($\lambda_\beta(\Omega)$): The first eigenvalue of the Laplacian with Robin conditions. Its variational characterization is:
   $$\lambda_\beta(\Omega) = \inf_{v \in H^1(\Omega)} \frac{\int_\Omega |\nabla v|^2 \, dx + \beta \int_{\partial\Omega} v^2 \, d\mathcal{H}^{n-1}}{\int_\Omega v^2 \, dx}$$

Motivation: While the Dirichlet case ($\beta \to \infty$) is well-studied (e.g., Saint-Venant and Faber-Krahn inequalities, Makai and Pólya bounds), the Robin case presents significant challenges. Unlike the Dirichlet case, Robin functionals do not exhibit monotonicity with respect to set inclusion, and standard scaling properties are altered. The authors aim to extend classical geometric inequalities (specifically those by Makai and Pólya) to the Robin setting and, crucially, to establish quantitative stability results. They seek to identify the optimal shapes (or sequences of shapes) that maximize or minimize these functionals relative to geometric quantities like measure ($|\Omega|$), perimeter ($P(\Omega)$), and inradius ($r(\Omega)$).

2. Methodology

The authors employ a combination of variational methods, geometric analysis, and one-dimensional reduction techniques:

• Distance Function Decomposition: A core technique involves decomposing the domain $\Omega$ based on the distance function from the boundary, $d(x, \partial\Omega)$. They utilize the Coarea Formula to transform integrals over $\Omega$ into integrals over the level sets of the distance function (inner parallel sets $\Omega_t$).
• One-Dimensional Reduction: The problem is reduced to analyzing a 1D eigenvalue problem on an interval $[0, s_0]$. By comparing the $n$-dimensional functionals to their 1D counterparts, the authors derive upper and lower bounds.
• Geometric Remainder Terms: To establish quantitative stability, the authors introduce specific "remainder terms" that measure how far a domain is from being optimal. The primary term used is:
  $$R(\Omega) = \frac{P(\Omega)r(\Omega)}{|\Omega|} - 1$$
  This term vanishes for sequences of "thinning" domains (slabs and cylinders). A secondary term, $A(\Omega) = \frac{w_\Omega}{\text{diam}(\Omega)}$ (ratio of minimal width to diameter), is also used to characterize thinness.
• Test Functions: The proofs rely on constructing specific test functions (often depending only on the distance to the boundary) to estimate the variational quotients for both $T_\beta$ and $\lambda_\beta$.

3. Key Contributions and Results

A. Upper Bounds and Generalized Makai Inequalities

The authors first establish an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function:
$$T_\beta(\Omega) \leq \int_\Omega d(x, \partial\Omega)^2 \, dx + \frac{2}{\beta} \int_\Omega d(x, \partial\Omega) \, dx$$
From this, they derive a Robin counterpart to the Makai inequality:
$$\frac{T_\beta(\Omega)}{r(\Omega)^2 |\Omega|} \leq \frac{1}{3} + \frac{1}{\beta r(\Omega)}$$
This generalizes the classical result for the Dirichlet case.

B. Quantitative Stability for Robin Torsion

The paper provides quantitative estimates showing how close the functionals are to their optimal values based on the geometry of the domain.

• Makai-type Quantitative Inequality: They prove that the difference between the normalized Robin Torsion and its theoretical upper bound is controlled by the remainder term $R(\Omega)$:
  $$C_1 R(\Omega) \leq \left(\frac{1}{3} + \frac{1}{\beta r(\Omega)}\right) - \frac{T_\beta(\Omega)}{r(\Omega)^2 |\Omega|} \leq C_2 R(\Omega)$$
  This confirms that slab-like domains (sequences of domains becoming infinitely thin in one direction) are the optimal sequences for this inequality.

• Pólya-type Quantitative Inequality: Similarly, for the Pólya functional involving perimeter and measure:
  $$C_3 R(\Omega)^3 \leq \frac{T_\beta(\Omega) P^2(\Omega)}{|\Omega|^3} - \left(\frac{1}{3} + \frac{1}{\beta r(\Omega)}\right) \leq C_4 R(\Omega)$$
  This establishes the sharpness of the lower bound for slab-like domains.

C. Quantitative Stability for Robin Eigenvalue

The authors address the upper bound for the first Robin eigenvalue, which is related to the 1D eigenvalue $\nu_1(\beta, s_0)$ where $s_0 = |\Omega|/P(\Omega)$.

• They prove a quantitative version of the inequality $\lambda_\beta(\Omega) \leq \nu_1(\beta, s_0)$:
  $$K_2 R(\Omega)^4 \leq \nu_1(\beta, s_0) - \lambda_\beta(\Omega) \leq K_1 R(\Omega)$$
  This result demonstrates that the gap between the actual eigenvalue and the 1D bound is strictly controlled by the "thinness" of the domain, again identifying slab-like domains as the optimizers.

D. Refinement using Asymmetry $A(\Omega)$

In the final section, the authors refine the lower bounds for the Pólya and eigenvalue functionals using the asymmetry measure $A(\Omega)$. They prove that the lower bounds can be improved to be linear in $A(\Omega)$ (or $A(\Omega)$ raised to a sharp power), providing a more precise geometric characterization of the deviation from optimality.

4. Significance and Implications

• Extension of Classical Theory: The paper successfully bridges the gap between the well-understood Dirichlet case and the more complex Robin case, extending fundamental inequalities (Makai, Pólya, Hersch-Protter) to include the parameter $\beta$.
• Identification of Optimizers: A major contribution is the rigorous identification of slab-like domains (and thinning cylinders) as the optimal sequences for these Robin inequalities. This contrasts with the Dirichlet case where balls are often the unique optimizers for volume-constrained problems; here, the lack of monotonicity leads to different extremal behaviors.
• Quantitative Stability: By providing explicit remainder terms ($R(\Omega)$ and $A(\Omega)$) with sharp exponents, the paper moves beyond qualitative inequalities. It quantifies how much the functional deviates from the optimal value based on the geometric "thinness" of the set. This is crucial for applications in shape optimization and stability analysis.
• Methodological Innovation: The use of the distance function combined with 1D reduction and careful estimation of inner parallel sets offers a robust framework for analyzing Robin boundary value problems on convex domains.

In summary, this work provides a comprehensive geometric analysis of the Robin Torsion and Eigenvalue, establishing sharp bounds and quantitative stability results that characterize the optimal shapes as sequences of thinning domains, thereby deepening the understanding of spectral geometry under Robin boundary conditions.