An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

1. Problem Statement

The paper investigates a shape optimization problem concerning the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains in $\mathbb{R}^m$.

• The Operator: The problem is defined by the weighted eigenvalue equation:
  $$\begin{cases} -\text{div}(e^{h(|x|)}\nabla u) = \lambda e^{h(|x|)}u, & x \in \Omega, \\ \frac{\partial u}{\partial \nu} + \alpha u = 0, & x \in \partial \Omega, \end{cases}$$
  where $\Omega$ is a bounded domain, $\nu$ is the outward unit normal, $\alpha \in \mathbb{R}$ is the Robin parameter, and $h(r)$ is a $C^2$ function on $[0, \infty)$.
• The Measure: The associated weighted measure is $d\gamma_h = e^{h(|x|)} dx$.
• The Objective: The authors seek to maximize the second eigenvalue $\lambda_{2,\alpha}(\Omega; \gamma_h)$ among all bounded Lipschitz domains $\Omega$ that are symmetric about the origin and have a prescribed weighted volume ($\gamma_h(\Omega) = \text{const}$).
• Context: This extends the classical Szegö-Weinberger inequality (which states that the ball maximizes the second Neumann eigenvalue) to the weighted setting with Robin boundary conditions.

2. Methodology

The proof relies on a combination of variational methods, spectral analysis on balls, and Weinberger's trick (constructing specific test functions based on the geometry of the ball).

A. Analysis on the Ball

The authors first analyze the eigenvalue problem on a round ball $B(R)$ centered at the origin.

• Separation of Variables: They decompose eigenfunctions into radial and spherical parts. They establish that the first eigenfunction is radial, while the second eigenfunctions are of the form $u_i(x) = g(r)\frac{x_i}{r}$, where $g(r)$ satisfies a specific radial ODE.
• Monotonicity Properties: A crucial step involves proving properties of the radial function $g(r)$ associated with the second eigenvalue. Specifically, they show that for $\alpha \leq 0$, $g(r)$ is strictly increasing and satisfies specific boundary inequalities.
• Steklov Connection: They relate the Robin problem to the Steklov eigenvalue problem (where $\lambda=0$ in the interior). They derive an upper bound for the first nonzero Steklov eigenvalue $\sigma_1(B; \gamma_h)$ on the ball, showing $\sigma_1(B; \gamma_h) \leq 1/R$ under the condition $h' \geq 0$.

B. Construction of Test Functions (Weinberger's Trick)

To prove the inequality for an arbitrary domain $\Omega$, the authors construct test functions $v_i(x) = g(|x|)\frac{x_i}{|x|}$ for $i=1, \dots, m$, where $g(r)$ is the radial part of the second eigenfunction on the ball $B$ (extended to infinity).

• Orthogonality: Due to the origin symmetry of $\Omega$ and the weighted measure, these test functions are orthogonal to the first eigenfunction $u_1$ of $\Omega$. This allows them to be used as valid test functions for the variational characterization of $\lambda_{2,\alpha}(\Omega)$.
• Variational Characterization: They substitute these test functions into the Rayleigh quotient for $\lambda_{2,\alpha}(\Omega)$.

C. Rearrangement and Monotonicity Lemmas

The core of the inequality proof involves comparing integrals over $\Omega$ and the ball $B$:

1. Definition of an Auxiliary Function: They define a function $F(r)$ involving $g(r)$, its derivative, and the parameters $\alpha$ and $h$.
2. Monotonicity of $F(r)$: They prove that under the assumptions $h' \geq 0$, $h'' \geq 0$, and $\alpha \in [-\sigma_1(B), 0]$, the function $F(r)$ is monotonically decreasing.
3. Hardy-Littlewood Inequality: Using the fact that $g(r)$ is increasing and $F(r)$ is decreasing, they apply a weighted rearrangement inequality (Lemma 3.2). This allows them to bound the integrals over $\Omega$ by integrals over the ball $B$ with the same weighted volume:
   $$\int_\Omega F(|x|) d\gamma_h \leq \int_B F(|x|) d\gamma_h \quad \text{and} \quad \int_\Omega g(|x|)^2 d\gamma_h \geq \int_B g(|x|)^2 d\gamma_h.$$

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Let $\Omega \subset \mathbb{R}^m$ be a bounded Lipschitz domain symmetric about the origin. Let $h(r) \in C^2([0, \infty))$ satisfy $h'(r) \geq 0$ and $h''(r) \geq 0$. Let $B$ be the origin-centered ball with the same weighted volume as $\Omega$.
If the Robin parameter $\alpha$ satisfies:
$$\alpha \in [-\sigma_1(B; \gamma_h), 0]$$
then:
$$\lambda_{2,\alpha}(\Omega; \gamma_h) \leq \lambda_{2,\alpha}(B; \gamma_h)$$
Equality holds if and only if $\Omega$ is isometric to the ball $B$.

Corollary 1.2 (Brock-Weinstock Inequality)

By setting $\alpha = -\sigma_1(B; \gamma_h)$, the authors recover the inequality for the first nonzero Steklov eigenvalue:
$$\sigma_1(\Omega; \gamma_h) \leq \sigma_1(B; \gamma_h)$$
This confirms and generalizes previous results by Brock, Chiacchio, and Mao-Zhang to the weighted setting with general $h(r)$.

Special Cases

• Neumann Case ($\alpha = 0$): The result implies that the ball maximizes the second Neumann eigenvalue for origin-symmetric domains with fixed weighted measure. This recovers a result by Brock, Chiacchio, and di Blasio.
• Gaussian Measure: The result applies to the specific case $h(r) = |x|^2/2$ (related to the Ornstein-Uhlenbeck operator), providing new insights into eigenvalue problems for $-\Delta u - x \cdot \nabla u = \lambda u$.

4. Significance

1. Extension of Classical Inequalities: The paper successfully extends the famous Szegö-Weinberger inequality from the standard Laplacian to the weighted Laplacian and from Neumann to Robin boundary conditions.
2. Handling Negative Robin Parameters: A significant technical achievement is handling the range of negative Robin parameters ($\alpha < 0$). While positive $\alpha$ often leads to different behaviors, the authors identify a specific range $[-\sigma_1(B), 0]$ where the ball remains the maximizer.
3. Weighted Measure Generality: The results hold for a broad class of radial weights $e^{h(|x|)}$ where $h$ is convex and non-decreasing, covering Gaussian measures and other important physical models.
4. Methodological Rigor: The proof provides a robust framework for analyzing weighted eigenvalue problems using separation of variables on balls and careful monotonicity analysis of the resulting radial functions, which can be adapted to other spectral optimization problems.

In summary, this paper establishes that for a specific class of weights and negative Robin parameters, the ball centered at the origin is the unique shape optimizer that maximizes the second Robin eigenvalue among all origin-symmetric domains of fixed weighted volume.