Local stability for a class of Saint-Venant type inequalities

Technical Summary: Local Stability for a Class of Saint-Venant Type Inequalities

Problem Statement
The paper addresses the quantitative stability of shape functionals of the form $J(\Omega) = \int_\Omega j(u) \, dx$, where $u$ is the solution to the Dirichlet torsion problem $-\Delta u = 1$ in $\Omega$ with $u=0$ on $\partial\Omega$. The function $j: \mathbb{R} \to \mathbb{R}$ is assumed to be convex, nonnegative, Lipschitz continuous, with $j(0)=0$.

By Talenti's comparison principle, it is known that among all domains of a fixed measure, the ball $\Omega^\sharp$ maximizes $J(\Omega)$. The central problem is to establish a sharp quantitative stability estimate for this extremal property. Specifically, the authors seek to prove that the "deficit" $J(\Omega^\sharp) - J(\Omega)$ controls the square of the distance of the domain $\Omega$ from spherical symmetry. While such results were previously established for the specific case of torsional rigidity (where $j(s)=s$) by Brasco, De Philippis, and Velichkov [9], this work extends the analysis to the broader class of functionals defined by general convex $j$.

Methodology
The analysis is restricted to the class of nearly spherical sets. A domain $\Omega$ is considered nearly spherical if its boundary is a small $C^{2,\gamma}$ normal perturbation of the unit sphere $B_1$, parametrized by a function $\phi$. The volume is preserved ($|\Omega| = |B_1|$), and the center of mass is fixed at the origin.

The primary tool employed is the method of shape derivatives. The authors consider a volume-preserving deformation flow $\{\Omega(t)\}_{t \ge 0}$ generated by an autonomous vector field $V$, starting from the unit ball. The stability is investigated by analyzing the first and second variations of the functional $f(t) = J(\Omega(t))$.

Key technical steps include:

1. First Variation: Using Hadamard's formula and the divergence-free property of the deformation field, the first derivative is computed. It is shown that $f'(0) = 0$, confirming the ball is a critical point.
2. Second Variation and the Adjoint State: The computation of the second derivative $f''(t)$ presents a challenge because it involves the material derivative of the state function, $\dot{u}$, which is difficult to control directly. To circumvent this, the authors introduce an adjoint state $p$, defined as the solution to an auxiliary boundary value problem:
   $$\begin{cases} -\Delta p = j'(u) & \text{in } \Omega \\ p = 0 & \text{on } \partial\Omega \end{cases}$$
   This technique allows the elimination of $\dot{u}$ from the boundary integral expressions of the second variation, yielding a tractable formula that depends only on the boundary perturbation and the gradients of $u$ and $p$.
3. Spectral Analysis: The second variation at $t=0$ is expressed as a quadratic form on the space of boundary perturbations. By expanding the perturbation in spherical harmonics, the authors analyze the coercivity of this form. They demonstrate that the quadratic form is positive definite on the subspace of perturbations orthogonal to the first-order spherical harmonics (which correspond to translations and volume changes, already fixed by the constraints).
4. Continuity and Coercivity: The proof relies on two main analytical estimates:
   • Improved Continuity: Showing that the second variation $f''(t)$ is close to $f''(0)$ for small perturbations, ensuring the nonlinear remainder does not destroy the positivity.
   • Coercivity: Proving that the second variation at the ball is bounded below by a constant times the squared $H^{1/2}$-norm of the perturbation, provided the perturbation is small and satisfies the volume/center constraints.

Key Contributions and Results
The main result is Theorem 1.2, which establishes a local stability inequality for nearly spherical sets. Specifically, there exist constants $\delta_c > 0$ and $C(n, j) > 0$ such that if $\Omega$ is a nearly spherical set of class $C^{2,\gamma}$ with $|\Omega| = |B_1|$ and center of mass at the origin, then:
$$J(B_1) - J(\Omega) \ge C(n, j) \|\phi\|_{H^{1/2}(\partial B_1)}^2$$
where $\phi$ is the parametrization of the boundary perturbation.

The paper highlights several specific applications where this general result applies:

• Torsional Rigidity: Recovering the result of [9] for $j(s)=s$.
• $L^p$-norms: Valid for $j(s) = s^p$ with $p \ge 2$.
• Moser–Trudinger Functional: Valid in dimension $n=2$ for $j(s) = e^{s^2} - 1$.

The authors also provide an Alternative Proof in Appendix B, establishing coercivity under weaker regularity assumptions on $j''$ (specifically $j'' \in L^{n/2}$) and specific structural conditions on $j$ and its derivatives.

Significance and Claims
The paper claims to extend the sharp quantitative stability theory from the specific case of torsional rigidity to a broad class of Saint-Venant type inequalities. The significance lies in the structural adaptation of the shape derivative method: by introducing the adjoint state, the authors successfully manage the additional terms arising from the nonlinearity of $j$ in the second variation.

The authors emphasize that while the result is local (restricted to nearly spherical sets), the use of the $H^{1/2}$-norm is crucial. This norm captures both volumetric deviations and oscillatory behavior of the boundary, providing a robust measure of asymmetry. The work builds directly upon the foundational quantitative stability results for the Faber–Krahn inequality [9] and the Saint-Venant inequality, demonstrating that the stability mechanism is robust across different convex functionals of the torsion function.