[Paper Review] An isoperimetric inequality for extremal Sobolev functions

Tom Carroll, Jesse Ratzkin|arXiv (Cornell University)|Aug 7, 2012

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes a reverse-Hölder inequality for extremal functions of the sharp Sobolev constant in bounded domains with Lipschitz boundaries, proving that the $ L^{p-1} $-norm of the extremal function is bounded below in terms of its $ L^p $-norm, with equality if and only if the domain is a ball. The result generalizes Payne and Rayner's inequality for eigenfunctions and relies on co-area formula, symmetrization, and one-dimensional variational analysis to derive a sharp isoperimetric-type inequality for Sobolev extremals.

ABSTRACT

Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1

Motivation & Objective

• To establish a reverse-Hölder inequality for extremal functions of the sharp Sobolev constant in bounded domains with Lipschitz boundaries.
• To generalize Payne and Rayner's isoperimetric inequality for Laplacian eigenfunctions to the broader class of extremal Sobolev functions.
• To characterize the equality case in the inequality, showing it holds if and only if the domain is a ball.
• To derive a sharp lower bound for the $ L^{p-1} $-norm of the extremal function in terms of its $ L^p $-norm and the Sobolev constant.

Proposed method

• Uses the co-area formula to define the auxiliary function $ H(t) = \int_{D_t} \phi^{p-1} \, d\mu $, where $ D_t = \{x \in D : \phi(x) > t\} $.
• Applies the co-area formula and divergence theorem to relate the derivative of $ H $ with respect to the volume $ V(t) = |D_t| $, yielding $ \frac{dH}{dV} = t^{p-1} $.
• Derives a differential inequality for $ H(V) $ using the Cauchy-Schwarz inequality and the PDE $ \Delta\phi + \lambda \phi^{p-1} = 0 $, leading to a lower bound on $ \frac{d^2H}{dV^2} $.
• Reduces the problem to a one-dimensional radial eigenvalue problem on a ball, analyzing the minimization of the Rayleigh quotient involving $ \rho^{1-n} f^2 $ and $ \rho^{1-n} f'^{p/(p-1)} $.
• Uses symmetrization and comparison with the ball domain $ D^* $ of equal volume to compare the Sobolev constant $ \mathcal{C}_p(D) $ with $ \mathcal{C}_p(D^*) $.
• Applies the Euler-Lagrange equation for the radial minimizer and derives a sharp bound via the generalized quotient $ \Lambda_* $, linking it to $ \mathcal{C}_p(D^*) $.

Experimental results

Research questions

• RQ1 Under what conditions does the $ L^{p-1} $-norm of an extremal Sobolev function dominate its $ L^p $-norm in a reverse-Hölder sense?
• RQ2 Can the Payne-Rayner inequality for Laplacian eigenfunctions be generalized to extremal functions of the sharp Sobolev constant for $ p \in (1, \frac{2n}{n-2}) $?
• RQ3 When does equality hold in the proposed reverse-Hölder inequality, and what geometric condition on the domain forces it?
• RQ4 How does the sharp Sobolev constant $ \mathcal{C}_p(D) $ relate to the geometry of the domain, particularly in comparison to the ball?

Key findings

• The paper establishes a reverse-Hölder inequality: $ \left(\int_D \phi^{p-1} \, d\mu\right)^2 \geq |D|^{\frac{n-2}{n}} \left(\int_D \phi^p \, d\mu\right)^{\frac{2(p-1)}{p}} \left[ \frac{2n^2 \omega_n^{2/n}}{p \, \mathcal{C}_p(D)} - (n-2) \frac{n \, \omega_n^{\frac{2}{n} + \frac{p^2 - p + 2}{p(p-1)}}}{\mathcal{C}_p(D^*)} \right] $.
• Equality holds in the inequality if and only if the domain $ D $ is a ball, and the extremal function $ \phi $ is radially symmetric.
• The inequality reduces to the Payne-Rayner result when $ p = 2 $, and to the reverse-Hölder inequality of [2] when $ n = 2 $.
• The sharp constant $ \mathcal{C}_p(D^*) $ for the ball satisfies $ \mathcal{C}_p(D^*) \leq (n\omega_n)^{\frac{p-2}{p}} \Lambda_* $, where $ \Lambda_* $ is the minimizer of a one-dimensional Rayleigh quotient.
• The proof relies on transforming the problem into a radial setting via symmetrization and using the co-area formula to relate level sets to the volume function $ V(t) $.
• The analysis shows that the $ L^{p-1} $-norm of $ \phi $ is minimized relative to its $ L^p $-norm when $ D $ is a ball, reflecting a geometric extremality.