Past ASK seminars

Abstract: In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when the steady states of the Keller–Segel model are considered with nonlocal chemical concentration diffusion. There are different possible notions of "non-local Neumann conditions" in the fractional setting. We will study the system both with spectral conditions, as was done by Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci) as was done by Cinti and Colasuonno. While for the spectral boundary condition there exists an extension theorem analogous to that of Caffarelli and Silvestre, such a theorem does not exist for integral boundary conditions. This makes it more difficult to obtain Liouville-type theorems and to show that the system has non-trivial positive solutions if the concentration parameter is big enough. Based on a joint project with Eleonora Cinti.

Abstract: We study the numerical approximation of degenerate Langevin-type Stochastic Partial Differential Equations (SPDEs) in two spatial dimensions. A semi-implicit Milstein finite difference scheme is proposed to manage the mixed deterministic-stochastic structure and the degeneracy of the operator. We perform a detailed mean-square stability and convergence analysis via Fourier techniques, providing explicit conditions for stability and convergence rates. To address the computational challenges typical of SPDE simulations, we also develop a Multi-level Monte Carlo (MLMC) framework coupled with the proposed scheme and analyze its theoretical complexity. This work is part of an ongoing study, with future developments focusing on implementation and the validation of these theoretical results through numerical experiments. 

Abstract:  µ-ellipticity describes certain degenerate forms of ellipticity typical of convex integrals at linear or nearly linear growth, such as the area integral or the iterated logarithmic model. The regularity of solutions to autonomous or totally differentiable problems is classical after Bombieri, De Giorgi and Miranda, Ladyzhenskaya and Ural’tseva and Frehse and Seregin. The anisotropic case is a later achievement of Bildhauer, Fuchs and Mingione, Beck and Schmidt and Gmeineder and Kristensen. However, all the approaches developed so far break down in presence of nondifferentiable ingredients. In particular, Schauder theory for certain significant anisotropic, nonautonomous functionals with Hölder continuous coefficients was only recently obtained by C. De Filippis and Mingione. We will see the validity of Schauder theory for anisotropic problems whose growth is arbitrarily close to linear within the maximal nonuniformity range, and discuss sharp results and insights on more general nonautonomous area type integrals. From a recent, joint work with Cristiana De Filippis (Parma) and Mirco Piccinini (Pisa)

Abstract:  We study a class of stochastic optimal control problems with infinite horizon and unbounded control operator using the dynamic programming principle. Our approach is related to the regularizing properties of the Ornstein-Uhlenbeck semigroup in separable Hilbert spaces. By applying a partial smoothing result, we prove the existence and uniqueness of solutions to the associated Hamilton-Jacobi-Bellman equations. These results can be applied to the analysis of  boundary control problems and stochastic delay equations. We will see how to provide an infinite dimensional reformulation of such problems and we will try to solve them by applying our abstract techniques.

Abstract Guido Drei

Abstract:

In this talk, I will discuss viscosity notions for minimal surfaces and how to use them to obtain regularity results. 

Abstract: The purpose of the talk is to gently introduce the audience to the regularity theory related to elliptic transmission problems. Transmission problems originally arised in the classical elasticity theory but soon have been used to model problems of different nature such as diffraction problems. We will show some properties of the solutions of such problems and some recent results on their optimal regularity when the transmission interface is C^{1,α} in the case of the Laplacian. In the final part of the talk we will present some results in the non-divergence case and some generalizations obtained in a recent work with D. Jesus (UNIBO)

AbstractFedericoFerri 

Abstract_ASK_D'Emilio 

Abstract_ASK_Tramontana

Abstract_ASK_Guarnotta

Abstract_ASK_Jesus

Abstract_ASK_Bozzola 

Abstract - We will introduce the evolutionary p-Laplacian operator, as the gradient flow of a suitable p-energy. On this focus, we will talk about existence/uniqueness and qualitative properties of the flow. 

Abstract_ASK_Cavina

Abstract_ASK_Circelli 

Many physical phenomena feature interfaces that minimize their surface area, for example, as the effects of surface tension. Within this framework, minimal surfaces and isoperimetric problems are predominantly practical phenomenological models. Among other motivations (such as image processing), there are also relevant phenomena presenting long-range interactions. Since their introduction in 2010 by Caffarelli, Roquejoffre and Savin, nonlocal minimal surfaces have attracted much attention, foremost to understand their regularity. We will describe the state of the art in this field, presenting the results obtained up to date, as well as the remaining open problems. In a very gentle manner, we will discuss the intriguing implications concerning minimal surfaces and isoperimetric problems, highlighting the remarkable resemblance and differences with the local counterparts.

We present an overview on regularity estimates for the solution of degenerate Kolmogorov equations with rough coefficients, namely Hölder continuous in space (and possibly only measurable in time). We focus our attention on recent results on the optimal regularity that can be obtained in this framework, in particular for the fundamental solution and Schauder estimates. The approach is based on the intrinsic geometry that the operator naturally induces on the space.

This is based on a joint work with Stefano Pagliarani and Andrea Pascucci.

Abstract_ASK_Ambrogi 

In the first part of the talk we will provide a gentle introduction to Carnot groups. Then, we present some recent results about a way of defining suitable fractional powers of the sub-Laplacian on an arbitrary Carnot group through an analytic continuation approach introduced by Landkof in Euclidean spaces. Furthermore, we present a stronger outcome in the setting of the Heisenberg group, which is the simplest non-commutative stratified group.

Eventually, depending on the time available, in this context we propose a geometrical application of our result. The research results of this talk were obtained in collaboration with F. Ferrari (University of Bologna).