Convex duality for principal frequencies

<abstract><p>We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 &lt; q &lt; 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).</p></abstract>

A short note on Helmholtz decompositions for bounded domains in $\mathbb{R}^3$

In this short note we consider several widely used $\mathsf {L}^{2}$-orthogonal Helmholtz decompositions for bounded domains in $\mathbb {R}^3$. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine this global property into a local equivalent: we show that functions from these spaces have zero mean in every subdomain of specific decompositions of the domain. An application of the zero mean properties is presented for convex domains. We introduce a specialized Poincaré-type inequality, and estimate the related unknown constant from above. The upper bound is derived using the upper bound for the Poincaré constant proven by Payne and Weinberger. This is then used to obtain a small improvement of upper bounds of two Maxwell-type constants originally proven by Pauly. Although the two dimensional case is not considered, all derived results can be repeated in $\mathbb {R}^2$ by similar calculations.

Poincaré and Log–Sobolev Inequalities for Mixtures

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

Logarithmic Sobolev inequalities for Moebius measures on spheres

In this paper, using the method in [1], i.e., reduce Moebius measures {\mu_{x}^{n}} indexed by {|x|<1} on spheres {S^{n-1}} ( {n\geq 3} ) to one-dimensional diffusions on {[0,\pi]} , we obtain that the optimal Poincaré constant is not greater than {\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by {C_{\rm LS}(\mu_{x}^{n})} behaves like {\frac{1}{n}\log(1+\frac{1}{1{-}|x|})} . As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than {L^{2}} -transportation-information inequalities.