Technical Summary: On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue, and Perimeter

1. Problem Statement

This paper investigates the interplay between two fundamental geometric quantities associated with the Laplacian on a domain $\Omega \subset \mathbb{R}^m$: the torsional rigidity $T(\Omega)$ and the first Dirichlet eigenvalue $\Lambda(\Omega)$. While the Saint–Venant inequality establishes that balls maximize torsional rigidity for a fixed volume (or perimeter), and the Faber–Krahn inequality establishes that balls minimize the first eigenvalue for a fixed volume (or perimeter), the behavior of their product is non-trivial due to these opposing extremal properties.

The primary objective is to optimize the product $\Lambda(\Omega)T(\Omega)$ under a prescribed perimeter constraint. The authors define the scale-invariant functional:
$$G(\Omega) := \frac{T(\Omega)\Lambda(\Omega)}{P(\Omega)^{\frac{m}{m-1}}}$$
where $P(\Omega)$ denotes the De Giorgi perimeter. The study is conducted across three distinct classes of domains:

1. $\mathcal{O}$: The class of open subsets of $\mathbb{R}^m$ with finite perimeter.
2. $\mathcal{C}$: The subclass of open, bounded, and convex sets.
3. $\mathcal{S}_{\delta, \gamma}$: The class of "nearly spherical" sets (perturbations of the unit ball $B_1$).

Additionally, the paper generalizes this problem to a family of functionals $F_q(\Omega) = T(\Omega)^q \Lambda(\Omega) V(\Omega)^{-\alpha_q}$ (where $\alpha_q = \frac{(m+2)q-2}{m}$) and a corresponding perimeter-constrained generalization $G_q(\Omega)$.

2. Methodology

The authors employ a combination of shape optimization theory, spectral analysis, and geometric inequalities.

• Shape Derivatives and Stability: The core analytical tool is the calculus of variations for shape functionals. The authors utilize the structure theorem for first and second shape derivatives (Theorem 2.6) to analyze the criticality and stability of the ball. They verify that the ball is a critical shape (first derivative vanishes) and investigate the coercivity of the second derivative (Hessian) in fractional Sobolev spaces ($H^s$).
• Spherical Harmonics Expansion: To analyze the second variation of the functionals on the ball, the authors expand boundary perturbations $h$ in terms of spherical harmonics. This allows them to compute the eigenvalues of the Hessian explicitly, determining whether the ball is a local minimizer, maximizer, or saddle point based on the sign of the coefficients $G_k$ or $F^q_k$.
• Homogenization and Homogenization Arguments: To establish bounds in the class of open sets ($\mathcal{O}$), the authors construct specific sequences of domains.
  • Minimizing sequences: Constructed using thinning domains or sequences where the isoperimetric ratio vanishes.
  • Maximizing sequences: Constructed via homogenization (introducing a dense array of small holes) to approach the supremum of the Pólya functional, combined with control of the isoperimetric ratio.
• Geometric Inequalities: The analysis relies heavily on known inequalities such as the Saint–Venant and Faber–Krahn inequalities, as well as specific bounds for convex sets involving diameter, inradius, and minimal width (e.g., Propositions 2.3–2.5).
• Regularity Analysis: For the convex case in the planar setting ($m=2$), a contradiction argument is used to prove that any maximizer must have a $C^1$ boundary, ruling out corners.

3. Key Contributions and Results

A. Optimization in the Class of Open Sets ($\mathcal{O}$)

• Infimum: The infimum of $G(\Omega)$ over $\mathcal{O}$ is 0. This is achieved by sequences of sets that become increasingly thin (vanishing isoperimetric ratio).
• Supremum: The supremum is not attained. It is given by the value $m^{\frac{m}{m-1}} \omega_m^{-\frac{1}{m-1}}$ (where $\omega_m$ is the volume of the unit ball). This value is approached by sequences constructed via homogenization of a ball, where the Pólya functional approaches 1 and the isoperimetric ratio is controlled.
• Generalization ($G_q$): For the generalized functional $G_q$, if $q \le \frac{2}{m+2}$, the ball is the global minimizer. If $q > \frac{2}{m+2}$, the infimum is 0. The supremum is $+\infty$ for $q < 1$ and finite for $q \ge 1$.

B. Optimization in the Class of Convex Sets ($\mathcal{C}$)

• Existence of Maximizers: Unlike the class $\mathcal{O}$, the supremum of $G(\Omega)$ over $\mathcal{C}$ is attained. The authors prove the existence of a maximizer by showing that maximizing sequences cannot "collapse" (i.e., the ratio of minimal width to diameter cannot vanish) and thus converge to a convex limit.
• Regularity: In the planar case ($m=2$), any maximizer of $G$ in $\mathcal{C}$ is of class $C^1$. The proof demonstrates that a corner point would allow for a perturbation that increases the functional value, contradicting maximality.
• The Ball as a Candidate: It remains an open problem whether the ball is the global maximizer in $\mathcal{C}$. Numerical simulations support this conjecture, but it is not proven.
• Generalization ($G_q$): The behavior of $G_q$ in $\mathcal{C}$ depends on $q$. For $q > 2/3$, the infimum is 0 and a maximizer exists. For $q < 2/3$, the supremum is $+\infty$.

C. Local Optimality and Stability (Nearly Spherical Domains)

The paper provides a detailed local analysis of the ball's stability for both $G$ and the family $F_q$ (and $G_q$) within the class of nearly spherical sets $\mathcal{S}_{\delta, \gamma}$.

• Local Maximality of the Ball for $G$: The ball is a strict local maximizer for the functional $G$. Specifically, there exists a constant $C > 0$ such that for any nearly spherical perturbation $B_h$:
  $$G(B_1) - G(B_h) > C \|h\|_{H^1(\partial B_1)}^2$$
• Local Behavior of $F_q$ and $G_q$: The stability of the ball depends critically on the parameter $q$:
  • $q < \frac{2}{m+2}$: The ball is a strict local minimizer.
  • $\frac{2}{m+2} < q < q^*$: The ball is a saddle point (neither a local max nor min). Here $q^*$ is a critical threshold involving the first zero of a Bessel function.
  • $q > q^*$: The ball becomes a strict local maximizer.
• Significance of $q^*$: The paper identifies an interval of $q$ values (specifically between $\frac{2}{m+2}$ and $q^*$) where the ball behaves as a saddle point. This contrasts with the global behavior where the ball is the global minimizer for $q \le \frac{2}{m+2}$.

4. Significance and Claims

The paper claims to resolve the optimization of the product of torsional rigidity and the first eigenvalue under a perimeter constraint, a problem that had not been fully characterized in the literature.

• Resolution of Extremal Values: The authors establish that while the infimum of the functional is 0 in the general class of open sets, the supremum is not attained. However, restricting the problem to convex sets ensures the existence of a maximizer.
• Local Stability: A significant contribution is the rigorous proof that the ball is a local maximizer for the perimeter-constrained product functional $G$. This provides strong theoretical support for the conjecture that the ball is the global maximizer in the convex class.
• Generalization: The work extends the classical Pólya functional analysis to a broader family of functionals $F_q$ and $G_q$, mapping out the transition from minimality to maximality (and saddle behavior) of the ball as the exponent $q$ varies.
• Methodological Rigor: The paper provides a complete framework for analyzing shape derivatives of mixed spectral-geometric functionals, specifically handling the interplay between volume, perimeter, and spectral quantities via spherical harmonic expansions.

The authors remain modest regarding the global optimality of the ball in the convex class, explicitly stating that whether the maximum is attained by a ball remains an open problem, despite numerical evidence and the proven local stability.