Seminario di Analisi Matematica

Il Seminario di Analisi Matematica si tiene presso il Dipartimento di Matematica "Federigo Enriques" dell'Università di Milano.

In questa pagina web è possibile trovare un elenco dei seminari e, scorrendo la pagina in basso, sono dati dettagli su aula, ora e abstract di ogni singolo seminario.

Per informazioni contattare giulio.ciraolo@unimi.it

Seminari A.A. 2019/2020

• Giovanni Comi - Fractional Sobolev and BV spaces: a distributional approach
• 23/01/2020: Paolo Giordano - A Grothendieck topos of generalized functions
• 09/01/2020: Luca Scarpa - From nonlocal to local phase-field models: asymptotic analysis and applications
• 11/12/2019: Dario Mazzoleni - Optimization results for the higher eigenvalues of the $p$-Laplacian
• 05/12/2019: Stefano Vita - Liouville type theorems and local regularity for degenerate or singular problems
• 27/11/2019: Cristiana De Filippis - Vectorial problems: sharp Lipschitz bounds and borderline regularity
• 20/11/2019: Alessandro Iacopetti - Spacelike hypersurfaces of prescribed mean curvature in the Lorentz-Minkowski space: existence, regularity and open problems
• 14/11/2019: Liliane Maia - Semilinear Parabolic Equations with asymptotically linear growth
• 17/10/2019: Nicola Garofalo - Sobolev and isoperimetric inequalities for Kolmogorov-Fokker-Planck operators
• 20/09/2019: Paul H. Rabinowitz - A Mountain Pass Theorem with Applications to Variational Gluing Problems
• 12/09/2019: Daisuke Naimen - Blow-up analysis for nodal radial solutions in Trudinger-Moser critical equations in R^2
• 12/09/2019: Hashizume Masato - Effect of compact term on maximization problem for Trudinger-Moser inequalities

Aule, Orari, Titoli e Abstracts

Abstract: Differently from their integer versions, the fractional Sobolev spaces $W^{\alpha, p}(\mathbb{R}^{n})$ do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature, in recent papers in collaboration with G. Stefani we introduce the new space $BV^{\alpha}(\mathbb{R}^{n})$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach. Thanks to the continuous inclusion $W^{\alpha, 1}(\mathbb{R}^{n}) \subset BV^{\alpha}(\mathbb{R}^{n})$, our theory provides a natural extension of the known fractional framework. In addition, we define in a similar way the distributional fractional Sobolev spaces $S^{\alpha, p}(\mathbb{R}^{n})$. In analogy with the classical $BV$ theory, we define sets with (locally) finite fractional Caccioppoli $\alpha$-perimeter and we partially extend De Giorgi's Blow-up Theorem to such sets, proving existence of blow-ups on points of the naturally defined fractional reduced boundary. In addition, we investigate the asymptotic behaviour of these fractional differential operators and we prove that the fractional $\alpha$-variation weakly and $\Gamma$-converges to the standard De Giorgi’s variation as $\alpha \to 1^{-}$, in perfect analogy with the well-known $\Gamma$-convergence result by Ambrosio-De Philippis-Martinazzi.

Abstract: We present a new approach to generalized functions, so-called generalized smooth functions (GSF). GSF are set-theoretical maps defined on, and taking values in the non-Archimedean ring (i.e. with real, infinite and infinitesimal numbers) of Robinson-Colombeau, and form a concrete category which unifies and extends Schwartz distributions and Colombeau generalized functions. The calculus of these generalized functions is closely related to classical analysis, with point values, the usual rules for differentiation and integration, free composition and hence non linear operations. We have classical theorems such as: intermediate value theorem, mean value theorems, extreme value theorem, several forms of Taylor formula, local and global inverse and implicit function theorems, a suitable sheaf property; Multidimensional integration with convergence theorems; A theory of non-Archimedean locally convex topological vector spaces of GSF; A theory of singular nonlinear ODE with Banach fixed point theorem, Picard-Lindelof theorem, maximal set of existence, Gronwall theorem, flux properties, continuous dependence on initial conditions, full compatibility with classical smooth solutions; Calculus of variations with: Euler-Lagrange equations, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. Using GSF, we can also prove a Picard-Lindelof theorem for nonlinear singular PDE in normal form. Therefore, all these results also apply to Schwartz distributions. Finally, we can define a concrete site and hence a Grothendieck topos of sheaves of generalized functions which contains the sheaves of Schwartz distributions and Colombeau generalized functions and all smooth manifolds. We hence present the planned future developments of this theory.

References

[1] A. Lecke, L. Luperi Baglini, P. Giordano, The classical theory of calculus of variations for generalized functions. Accepted in Advances in Nonlinear Analysis. DOI: 10.1515/anona-2017-0150.

[2] P. Giordano, M. Kunzinger, Inverse Function Theorems for Generalized Smooth Functions. Chapter in "Generalized Functions and Fourier Analysis", Volume 260 of the series Operator Theory: Advances and Applications pp. 95-114.

[3] P. Giordano, M. Kunzinger, H. Vernaeve, Strongly internal sets and generalized smooth functions. Journal of Mathematical Analysis and Applications, volume 422, issue 1, 2015, pp. 56-71.

[4] P. Giordano, M. Kunzinger, A convenient notion of compact set for generalized functions. Proceedings of the Edinburgh Mathematical Society, Volume 61, Issue 1, February 2018, pp. 57-92.

[5] P. Giordano, home page: www.mat.univie.ac.at/~giordap7/

Abstract: We give an overview of some recent results about the convergence of nonlocal Cahn-Hilliard systems to their respective local counterparts, as the convolution kernel approximates a Dirac delta. The double-well potential is allowed to be of logarithmic or double-obstacle type, and possible viscous and convective terms are included. The asymptotics is carried out by means of monotone analysis, variational techniques, and Gamma convergence results. Finally, applications to the study of more general nonlocal and local phase-field models are discussed.

This study is based on joint works with Elisa Davoli and Lara Trussardi (University of Vienna, Austria).

Abstract: In this talk we study the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence. This is a joint work with Marco Degiovanni.

Abstract: We consider a class of equations in divergence form with a weight which is singular or degenerate on a hyperplane, the characteristic manifold. We study local regularity of solutions, showing how boundary conditions on the characteristic manifold do affect such regularity. Our analysis relies in uniform bounds in Holder spaces for solutions to a class of uniformly elliptic approximating problems as the parameter epsilon of approximation tends to zero. Our method is based upon blow-up and appropriate Liouville type theorems.

Abstract: pdf.

Abstract: In this talk we present some results concerning the existence and the regularity of entire spacelike hypersurfaces of prescribed mean curvature in the Lorentz-Minkowski space $L^{N+1}$, satisfying a homogenous boundary condition at infinity. In particular we discuss the regularity of the minimizer of the Born-Infeld energy, when the given distribution is in $L^p$, for $p>N$.

In the second part we consider the case of generic bounded domains and show recent contributions and open problems concerning the Plateau problem for spacelike vertical graphs and radial graphs in $L^{N+1}$.

These results are obtained in collaboration with D. Bonheure (Université Libre de Bruxelles) and they are contained in the following two papers:

[1] Arch. Ration. Mech. Anal., Vol. 232 (2019);

[2] Analysis & PDE, Vol. 12 (2019).

Abstract: We present some recent work on the existence and behaviour of solutions for a class of semilinear parabolic equation, defined on a bounded smooth domain and we assume that the nonlinearity is asymptotically linear at infinity. We analyse the behavior of the solutions when the initial data varies in the phase space. We obtain global solutions which may be bounded or blowup in infinite time (grow-up). Our main tools are the comparison principle and variational methods. Particular attention is paid to initial data at high energy level. We use the Nehari manifold to separate the phase space into regions of initial data where uniform boundedness or grow- up behavior of the semiflow may occur. This is work in collaboration with Juliana Pimentel (UFRJ, Brazil).

Abstract: In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order hypoellipitc evolution equation which displays many challenging features. Thirty years later, in the opening of his famous 1967 hypoellipticity paper, Hormander discussed a general class of degenerate Ornstein-Uhlenbeck operators that includes Kolmogorov’s as a special case. The natural semigroups attached to such equations need not be symmetric or doubling, thus the existing theories do not readily apply. As a consequence, despite the large amount of work done by many people over the past thirty years, some basic questions have remained unsettled, such as Hardy-Littlewood-Sobolev and Isoperimetric inequalities and the study of local and nonlocal minimal surfaces. In this lecture I will present some interesting developments in the above program.

This is joint work with Giulio Tralli.

Abstract: The effect of non-degeneracy conditions on the applicability of variational gluing arguments for some variational problems possessing mountain pass structure will be discussed.

Abstract: We consider maximization problems on the Trudinger-Moser inequalities with compact terms. Existence of a maximizer depends on the growth of the compact term. In this talk, we show sufficiently conditions of existence and nonexistence on the compact term.

Abstract: We study the aymptotic behavior of low energy nodal radial solutions as the growth rate of the nonlinearity goes to a threshold between the existence and nonexistence of nodal radial solutions. The solution exhibits a multiple concentration behavior together with a convergence to the least energy solution of a critical problem. We prove that the asymptotic profile of each concentration part, with an appropriate scaling, is given by the classical Liouville equation in $\mathbb{R}^2$. This talk is based on a joint work with Massimo Grossi at Sapienza University of Rome.