Sharp quantitative Talenti's inequality in particular cases

1. Problem Statement

The paper investigates quantitative stability versions of Talenti's symmetrization inequality.

• Classical Context: Talenti's inequality states that for a bounded domain $\Omega \subset \mathbb{R}^n$ and a non-negative source term $f \in L^2(\Omega)$, the solution $u_f$ to the Dirichlet problem $-\Delta u_f = f$ satisfies a pointwise comparison with the solution $v$ to the symmetrized problem $-\Delta v = f^\sharp$ on the symmetrized domain $\Omega^\sharp$ (a ball of the same volume). Specifically, the Schwarz symmetrization of the solution satisfies $u_f^\sharp \le v$. Consequently, for any $p \in [1, \infty]$, $\|u_f\|_{L^p(\Omega)} \le \|v\|_{L^p(\Omega^\sharp)}$.
• The Gap: While the inequality is well-known, quantitative versions (stability inequalities) measuring how close the inequality is to equality based on the "asymmetry" of the domain and the function are less developed. Previous results provided stability estimates but with non-sharp exponents or without controlling the asymmetry of the source term $f$.
• Specific Focus: This paper focuses on the case where:
  1. The domain is fixed as the unit ball $B_1$ (so $\Omega = \Omega^\sharp$).
  2. The source term $f$ is the characteristic function of a subset $E \subset B_1$ (i.e., $f = \chi_E$).
  3. The goal is to establish a sharp stability inequality of the form:
     $$\|u_{B^*}\|_{L^p(B_1)} - \|u_E\|_{L^p(B_1)} \ge c |E \Delta B^*|^2$$
     where $B^*$ is the centered ball with the same volume as $E$, and $|E \Delta B^*|$ is the Fraenkel asymmetry (symmetric difference).

2. Methodology

The authors employ a combination of shape optimization, calculus of variations, and spectral analysis on the sphere. The strategy is divided based on the value of $p$:

A. Case $p = 1$

• Approach: Direct analysis using Talenti's inequality and linearity.
• Technique: The authors formulate a shape optimization problem to find the "worst-case" set $E$ for a fixed asymmetry $\delta = |E \Delta B^*|$. They identify that the optimal (worst) set is a specific annular configuration (a union of a ball and an annulus).
• Result: By explicitly computing the energy difference for this specific annular set, they derive the quadratic lower bound.

B. Case $1 < p < \infty$

• Framework: The problem is treated as a shape optimization problem maximizing the functional $J(E) = \int_{B_1} j(u_E) dx$ with $j(s) = s^p$.
• Selection Principle: The proof follows the "selection principle" strategy (inspired by Cicalese and Leonardi for the isoperimetric inequality).
  1. Reduction to Smooth Deformations: It is shown that proving the stability inequality for general sets reduces to proving it for smooth deformations of the ball $B^*$. This is achieved using a Quantitative Bathtub Principle (Theorem 3.3) to replace arbitrary sets with level sets of an adjoint state, which are smooth.
  2. Shape Derivatives: The authors compute the first and second-order shape derivatives of the functional $J$ with respect to domain deformations $\Phi$.
  3. Coercivity: They prove that the second-order derivative (the Hessian of the Lagrangian) is coercive (strictly negative definite) on the space of volume-preserving deformations orthogonal to the boundary. This involves:
     • Diagonalizing the associated operator using spherical harmonics.
     • Proving that the eigenvalues corresponding to non-constant modes are strictly bounded away from zero.
  4. Taylor Expansion: A refined Taylor expansion with a controlled remainder term (using improved continuity of the second derivative) establishes the quadratic stability.

C. Case $p = \infty$

• Challenge: The $L^\infty$ norm is not differentiable in the standard sense, and the maximum point of the solution moves with the domain.
• Innovation: The authors compute the second-order shape derivative of the $L^\infty$ norm. This requires:
  • Proving the differentiability of the maximum point $x_E$ of the solution $u_E$ with respect to domain deformations (using the Implicit Function Theorem and Schauder estimates).
  • Using the Green's function of the ball as the "adjoint state" to represent the solution and its derivatives.
  • Establishing coercivity and continuity for this specific norm.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

For $n \ge 1$, $p \in [1, \infty]$, and a fixed volume $m$, there exists a constant $c > 0$ such that for any measurable set $E \subset B_1$ with $|E|=m$:
$$\|u_{B^*}\|_{L^p(B_1)} - \|u_E\|_{L^p(B_1)} \ge c |E \Delta B^*|^2$$

• Sharpness: The exponent 2 is proven to be sharp. The authors construct a sequence of sets (specifically the annular sets $A_\delta$) where the deficit scales exactly as $\delta^2$, proving that no higher exponent (e.g., $2+\epsilon$) is possible.
• Explicit Constants: For $p=1$, the constant $c$ is explicitly calculated. For $p > 1$, the existence is proven via contradiction, so the constant is not explicit.

Theoretical Tools Developed

1. General Stability for Shape Functionals (Theorem 1.3): The paper proves a general stability result for functionals of the form $J(V) = \int j(u_V)$ where $j$ is strictly convex and increasing. This extends the result beyond just $L^p$ norms to a broader class of functionals.
2. Second-Order Shape Derivative of $L^\infty$: A novel computation of the second-order shape derivative for the $L^\infty$ norm of the solution to the Poisson equation, accounting for the movement of the maximum point.
3. Spectral Analysis of the Hessian: A detailed spectral analysis of the linearized operator around the ball, showing that the first non-trivial eigenvalue (associated with spherical harmonics of degree 1) is strictly separated from the critical threshold, ensuring coercivity.

4. Significance

• Sharpness: This is the first result to provide a sharp exponent (2) for the quantitative Talenti inequality in the context of characteristic functions. Previous results had suboptimal exponents (e.g., $2+p$ or $3$).
• Methodological Advancement: The paper successfully adapts the "selection principle" and shape derivative techniques to the specific constraints of the Talenti inequality, particularly handling the non-smoothness of the $L^\infty$ norm and the specific structure of characteristic functions.
• Foundation for Future Work: The techniques developed, particularly the handling of the $L^\infty$ case and the general convex functional framework, pave the way for investigating more general source terms $f$ (not just characteristic functions) and other domains in future research.
• Rigidity: The work reinforces the rigidity of Talenti's inequality, quantifying exactly how much the solution's norm degrades as the source set deviates from a ball.

In summary, the paper provides a rigorous, sharp quantitative refinement of a classical inequality in PDEs, utilizing advanced tools from shape optimization and spectral theory to resolve the stability exponent for characteristic functions on a ball.