On principal frequencies and inradius in convex sets

We generalize to the case of the $p-$Laplacian an old result by Hersch and Protter. Namely, we show that it is possible to estimate from below the first eigenvalue of the Dirichlet $p-$Laplacian of a convex set in terms of its inradius. We also prove a lower bound in terms of isoperimetric ratios and we briefly discuss the more general case of Poincar\'e-Sobolev embedding constants. Eventually, we highlight an open problem.Comment: 20 pages, 3 figure

A note on some Poincar\'e inequalities on convex sets by Optimal Transport methods

We show that a class of Poincar\'e-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly unbounded.Comment: 13 page

A continuous model of transportation revisited

We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J Control Optim 47: 1330-1350, 2008), which in turn is the continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov decomposition theorem for normal 1-currents.Comment: 26 pages. Theorem A.20 of v1 was not correct: we removed it and replaced it with the counterexample A.18 in v2. We also made some improvements to the wording and corrected some typo

The second eigenvalue of the fractional p−p-Laplacian

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.Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1<p<2$. We fixed this issu

Higher Sobolev regularity for the fractional p−p-Laplace equation in the superquadratic case

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.Comment: 36 page

A pathological example in Nonlinear Spectral Theory

We construct an open set $\Omega\subset\mathbb{R}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.Comment: 9 pages, 1 figur

Improved energy bounds for Schr\"odinger operators

Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative stability estimates.Comment: 31 page

Beyond Cheeger's constant

The Cheeger constant of an open set of the Euclidean space is defined by minimizing the ratio "perimeter over volume", among all its smooth compactly contained subsets. We consider a natural variant of this problem, where the volume of admissible sets is raised to any positive power. We show that for {\it sublinear} powers, all these generalized Cheeger constants are equivalent to the standard one, by means of a universal two-sided estimate. We also show that this equivalence breaks down for {\it superlinear} powers. In this case, some weird phenomena appear. We finally consider the case of convex planar sets and prove an existence result, under optimal assumptions.Comment: 33 pages, 2 figure

Spectral inequalities in quantitative form

We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian.Comment: 71 pages, 4 figures, 6 open problems, 76 references. This is a chapter of the forthcoming book "Shape Optimization and Spectral Theory", edited by Antoine Henrot and published by De Gruyte