Convegno
“RECENT PROGRESS IN GEOMETRIC ANALYSIS AND PDES”

Solutions of degenerate nonlinear diffusion equations such as the porous media equation have the property of finite speed of propagation of disturbances. In this talk we review a class of more degenerate equations characterized by vanishing speed of propagation. We explain how an example of such an equation arises as the singular limit of a parabolic equation describing the dynamics of glasslike fluids. This naturally leads to the problem of finding asymptotic formulas for the velocities of diffuse free boundaries. This is work in progress with Roberto Benzi (Dept. Physics, Univ. of Roma Tor Vergata) and Francesco Deangelis (GSSI, L'Aquila).

We shall present a summary of a forthcoming book with Daniel Hauer (University of Sydney). The aim of this monograph is to introduce natural and simple functional analytic methods to deduce $L^1-L^\infty$ regularization estimates on nonlinear semigroups from natural Gagliardo-Nirenberg inequalities satisfied by their infinitesimal generator. This enables one to treat in an optimal and unified way a wealth of examples, including the p-Laplace operator, the porous medium operator, as well as variations and combinations of them.

I will discuss firs the validity of the weak Maximum Principle (wMP) for vector functions u = (u1, .., um) satisfying systems of the form Au + Cu ≥ 0 in a bounded open set Ω of Rn where A is a diagonal matrix of linear degenerate second order elliptic operators and C is a cooperative matrix. Next some counterexamples to the validity of (wMP) are discussed when non diagonal couplings in first order partial derivatives of the ui appear in the system. In this more general setting I will show, through a suitable reduction to a nonlinear scalar equation of Bellman type, that some algebraic condition on the structure of gradient couplings and a cooperativity condition on the matrix of zero order couplings guarantee the existence of invariant cones in the sense of Weinberger.

We investigate the asymptotic behaviour, with respect to Gamma-convergence, of action functionals made up of a kinetic term and a term induced by the the gradient of a convex function. The initial motivation has been the derivation of continuous models from particle systems, but the result has an independent interest. Joint works with A.Baradat, Y.Brenier and C.Brena.

By using optimal mass transport theory we prove a sharp isoperimetric inequal- ity in CD(0, N ) metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals as time elapsed after infection. During the 70s, many mathematical studies where developed when equations are structured by age, size or a physiological trait. The talk will present some structured equations, show that a universal relative entropy structure is available in the linear case. In the nonlinear cases it might be that periodic solutions occur, which can be interpreted, e.g., as network activity in the neuroscience. When the equations are conservation laws, a variant of the Monge-Kantorovich distance also gives a general control of solutions.

Although the Laplacian cannot be inverted on L1, the exterior differential can sometimes be inverted on L1. This discovery, due to Bourgain, Brezis et al. in the early 2000's, can be explained in geometric terms. Such results have generalizations in a Heisenberg group setting. This is joint work with A. Baldi, B. Franchi and F. Tripaldi.

We prove uniform convergence in Lipschitz domains in $\mathbb{R}^n$ (and in $C^{1,1}$ domains in the Heisenberg group) of approximations to $p$-harmonic functions obtained using the natural $p$-means introduced by Ishiwata, Magnanini, and Wadade in 2017. This talk is based in joint work with Andr\'as Domokos and Diego Ricciotti (Sacramento) and Bianca Stroffolini (Naples)

This research originates from recent results by M. Goldman and F. Otto concerning regularity of optimal transport maps for the quadratic cost. We consider cost functions having the form c(x, y) = h(x − y), where h is positively homogeneous of degree p ≥ 2 and h ∈ C2(Rn). A mapping T : Rn → Rn is c-monotone if c(Tx,x) + c(Ty,y) ≤ c(Tx,y) + c(Ty,x). Using Green’s representation formulas, if T is c-monotone, we prove local L∞- estimates of Tx−x in terms of Lp-averages of Tx−x. From this we deduce estimates for the interpolating maps between T and Id, and when T is optimal, L∞-estimates of T −1x − x. As a consequence of the technique, we also obtain a.e. differentiability of monotone maps. This is joint work with Annamaria Montanari.

It is an ongoing project to study collections of measures that are negligible in a sense of ``modules". The idea is originated in complex analysis as ``a conformal module of a family of curves" in looking for an invariant object under conformal transformations on the complex plane. The notion is closely related to the potential theory, certain capacity, and Hausdorff measure. Later the definition of the module was successfully applied to the nonlinear potential theory and quasiconformal analysis in a wider sense in Euclidean spaces. B. Fuglede, by studying the completion of functional spaces, generalized the notion of the module of a family of curves to the module of a family of measures. The arc length of a curve was thought of as a measure. A collection of measures is exceptional if the corresponding module vanishes. In the talk, I will remind examples of exceptional measures in Euclidean space. We aim to find exceptional families of measures on Carnot groups, related to geometric objects such as "intrinsic graphs". It leads to the notion of a Grassmannian on specific Carnot groups.

Let $\mu$ be a measure concentrated on a domain $ D \subset \mathbb {R}^N$ , and let $ x_0 \in D$. Denote by $ \Gamma$ the fundamental solution of the Laplacian, and by $\Gamma_{\mu}$ the Newtonian potential of $\mu$. We say that $\mu$ is a polarity measure for $D$ at $x_0$ if $\Gamma_{\mu} = \Gamma (x_0 - x)$ for every $x$ in the complementary of $D$. If we also have $\Gamma_{\mu} < \Gamma (x_0 - x)$ for every $x \in D$ then we say that $\mu$ is a strong polarity measure for $D$ at $x_0$. In the present talk we first recall the following results: A. Every sufficiently smooth domain supports a polarity measure at an arbitrarily given point. B. Every strong polarity measure characterizes its supporting domain. Then we show how to extend A and B to the general context of the hypoellipitic semi-elliptic linear second order PDEs. All the results we present have been obtained in collaboration with Giovanni Cupini.

Suppose that, for a surface $S\subset\bR^n$ with (weighted) surface measure $\sigma$ and for some $p,q$ with $p\in(1,2)$, the Fourier restriction operator $\cR:f\longmapsto \widehat f_S$ satisfies the inequality $$ \|\mathcal R f\|_{L^q(S,\sigma)}\le C\|f\|_{L^p(\mathbb R^n)}\ ,\qquad \forall f\in\cS(\mathbb R^n)\ . $$ Then extendability of $\mathcal R$ to all of $L^p(\mathbb R^n)$ indicates, heuristically, that, for general $f\in L^p(\mathbb R^n)$, $\widehat f$ can be assigned values on $S$, despite the fact that it is only defined a.e. The notion of ``maximal restriction operator'' has been introduced in a paper of 2019 by D.~M\"uller, J.~Wright and myself, for the purpose of giving measure-theoretic ground to this statement. In this talk I give a precise presentation of the problem, the improvements of our original result by various authors and some of the open problems.

The problem of Holder regularity of a variational solutions u = u(x) of a degenerate uniformly elliptic second order equations as (1) \sum_{i=1}^n Di(w(x)Diu(x)) = 0; x\in\Omega\subset R^n has been addressed since the beginning of the seventies. Now it is well known that if w belongs to the Muckenhaupt class A_2 then variational solutions of (1) are Holder continuous. On the other side the necessity of the assumption w\in A_2, or of similar structural assumptions on the weights, is far from being well understood. The simpler question of the necessity/sufficiency of quantitative assumptions on w and 1/w, even if better understood, is not yet completely settled.