On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity

On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on $\partial \Omega$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(\Omega)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial \Omega$; by using a suitable $P$-function we prove that, if $\Omega$ satisfies the same assumptions as above and in addition contains a ball with touches $\partial \Omega$ at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of $\Omega$ coincide. In turn, in dimension $n=2$, this entails that $\Omega$ must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class $C^2$.Comment: 26 pages, 1 figur

The area measure of log-concave functions and related inequalities

On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.Comment: 36 page

On a geometric combination of functions related to Prekopa-Leindler inequality

We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Pr\'ekopa-Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the $0$-sum, we get an alternative proof of the log-Brunn-Minkowski inequality for unconditional convex bodies and for convex bodies with $n$ symmetries.Comment: 22 pages, 1 figur

Sensitivity of the Compliance and of the Wasserstein Distance with Respect to a Varying Source

We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as p→ + ∞, which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as p→ + ∞