Convex bodies: the brunn–minkowski theory

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.

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A User’s Guide to Optimal Transport

We prove sharp regularity results for a general class of functionals of the type $$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$
 featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1<p < q, \quad a(x)\ge 0, \end{aligned}$$
 with $$0<\nu \le b(\cdot )\le L $$
 . This changes its ellipticity rate according to the geometry of the level set $$\{a(x)=0\}$$
 of the modulating coefficient $$a(\cdot )$$
 . We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

Regularity for general functionals with double phase

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \[ \inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}. \] We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.

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The second eigenvalue of the fractional p-Laplacian

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known {\it Cheeger problem}. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.

https://www.ems-ph.org/journals/show_pdf.php?issn=1463-9963&vol=16&iss=3&rank=5

The fractional Cheeger problem

The second eigenvalue of the fractional $p-$Laplacian

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

https://projecteuclid.org/download/pdf_1/euclid.kmj/1414674621

Lorenzo Brasco

Papers

The second eigenvalue of the fractional p-Laplacian

The fractional Cheeger problem

The second eigenvalue of the fractional $p-$Laplacian

Convexity properties of Dirichlet integrals and Picone-type inequalities

Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case