Zachary Adams
“Quasi-ergodicity and Q-processes of parabolic SPDE“
We study the dynamics of a parabolic SPDE in a bounded region E of its phase space, under the assumption that the solution hits the boundary of E at some almost surely finite time. In particular, we discuss the existence and uniqueness of quasi-ergodic distributions (QED), which characterize the long time behaviour of the SPDE conditioned on remaining in E for all time. Following this, we present applications to the study of stochastic travelling waves in reaction-diffusion systems. Time permitting, we discuss functional inequalities for QED, and their consequences for the solution of an SPDE.

Davide A. Bignamini
“Schauder estimates for stationary and evolution equations associated with stochastic reaction-diffusion equations driven by colored noise“
We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of R^d for d=1,2,3. We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution equations driven by the weak generator of the corresponding transition semigroup.

Krzysztof Bogdan
“The p-forms (1 < p < ∞)”
I will discuss optimal Hardy inequality for the fractional Laplacian in the setting of L^p(R^d) and the related p-forms, or Bregman-Sobolev forms. I also plan to mention a wider research program involving the p-forms.
The following relevant papers can be found on arXiv:
Optimal Hardy inequality for the fractional Laplacian on L^p, 2021, K. Bogdan, T. Jakubowski, J. Lenczewska, K. Pietruska-Pałuba
Nonlinear nonlocal Douglas identity, 2020, K. Bogdan, T. Grzywny, K. Pietruska-Pałuba, A. Rutkowski

Giovanni M. Brigati
“Modified Poincaré inequalities and hypocoercivity”
Evolution equations of diffusion type often show relaxation of the initial data towards a equilibrium. A standard way to establish such results passes trough the entropy method. Indeed, the evolution of an entropy functional along the flow of the equation is compared with the first variation of the entropy itself. A functional inequality is needed at this point. The value of the constant in the aforementioned inequality reflects directly on the convergence rates of the solutions to the equation to equilibrium. The former is a variational problem in its nature. We review such a theory for the Ornstein-Uhlenbeck equation. Finally, we present new results in this spirit for the Vlasov-Ornstein-Uhlenbeck equation.

Daniela Di Donato
“A new point of view for intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano in metric spaces”
We introduce the notion of intrinsically Lipschitz graphs in metric spaces. We generalize the concept of intrinsically Lipschitz graphs given by Franchi, Serapioni and Serra Cassano in the context of subRiemannian Carnot groups. We do this by focusing our attention on the graphs property instead of the maps one. More precisely, we prove some relevant results in metric spaces as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for the so-called intrinsically Lipschitz sections. This is a joint work with Enrico Le Donne.

Silvia Frassu
“Five solutions for the fractional p-Laplacian with noncoercive energy“
In this talk we deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. We focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably truncated reactions, we show the existence of five nontrivial solutions: two positive, two negative, and one nodal. This work is in collaboration with Antonio Iannizzotto.
References:
S. Frassu, A. Iannizzotto, Five solutions for the fractional p-Laplacian with noncoercive energy, NoDEA Nonlinear Differential Equations Appl. 29, 2022. doi:10.1007/s00030-022-00777-0 N.S.
Papageorgiou, G. Smyrlis, Nonlinear elliptic equations with an asymptotically linear reaction term, Nonlinear Anal. 71, 3129–3151, 2009. doi:10.1016/j.na.2009.01.224

Megan Griffin-Pickering
“Mean field games with control on the acceleration: a variational approach“
The theory of mean field games aims to describe the limits of Nash equilibria for differential games as the number of players tends to infinity. If players control their state by choosing their acceleration, then the mean field games system describing this equilibrium includes a kinetic transport term. Previous results on the well-posedness theory of mean field games of this type assume either that the running and final costs are regularising functionals of the density variable, or the presence of noise – that is, a second-order system. I will present recently obtained results in which we construct global-in-time weak solutions for a deterministic `kinetic’ mean field game with local (hence non-regularising) couplings, under suitable convexity and monotonicity conditions. Our approach is based on a characterisation of the solutions through two optimisation problems in duality. Furthermore, under stronger monotonicity/convexity assumptions, we obtain Sobolev regularity estimates on the solutions. This talk is based on joint work with Alpár Mészáros.

Erica Ipocoana
“Higher fractional differentiability for solutions to a class of double-phase obstacle problems“
We aim to study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems with a double-phase functional.
Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property. The main difficulty here is that the functional depends both on the x-variable and the w-variable. In order to overcome this issue, we adapt the strategy presented in [1]. Namely, we pass to a “freezed” obstacle problem for whom the results proved in [2] hold, so that we have that the solutions to the “freezed” obstacle problem inherit some Besov regularity of the obstacle. This is a joint project with Antonio Giuseppe Grimaldi.
References:
[1] M. Eleuteri and A. Passarelli di Napoli, Regularity results for a class of non- differentiable obstacle problems Nonlinear Analysis 194, 111434 (2020)
[2] A.G. Grimaldi and E. Ipocoana, Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions, Adv. Calc. Var. (2022) DOI: 10.1515/acv-2021-0074.

Laura Kanzler
“Kinetic models for non-instantaneous binary collisions“
In this talk we introduce a new class of kinetic models, which overcome the standard assumption in kinetic transport theory that collision processes happen instantaneously. We investigate a kinetic model with non-instantaneous binary alignment collisions between particles. The collisions are described by a transport process in the joint state space of a pair of particles, where the states of the particles approach their midpoint. For two spatially homogeneous models with deterministic or stochastic collision times existence and uniqueness of solutions, the long time behavior, and the instantaneous limit are considered, where the latter leads to standard kinetic models of Boltzmann type.

Erik Lindgren
“Nonlinear nonlocal equations”
In this talk, I will discuss some classes of nonlocal or fractional partial differential equations. In particular, I will describe recent developments for fractional versions of nonlinear equations such as the p-Laplace equation. Focus will be on regularity theory.

Amélie J. Loher
“Quantitative De Giorgi Methods in Kinetic Theory for Non-Local Operators “
We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker- Planck type equations with a non-local diffusion operator for the full range of the non-locality exponent $s \in (0, 1)$. This implies Hölder continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

Mirco Piccinini
“Nonlocal Harnack inequalities in the Heisenberg group Hⁿ“
In the non-Euclidean setting of the Heisenberg group Hⁿ we present some qualitative properties of the solution u to the Dirichlet problem
(1) Lu=f, in Ω⊂ Hⁿ, u = g, in Hⁿ \ Ω,
where f ≡ f(·,u) ∈ L^∞_loc(Hⁿ) , g ∈ W^s,p(Hⁿ) and L is an integro-differential operator of differentiability order s ∈ (0,1) and summability growth p ∈ (1,∞), whose prototype is the standard classical fractional sub Laplacian in the Heisenberg group. We show that weak solutions to Problem (1) satisfies some Harnack-type inequalities, extending to the Heisenberg framework the results of [1,2], and in turn generalizing them to the non homogeneous case. Moreover, in the linear case when p = 2, we state a robustness result of these estimates in the limit as the differentiability exponent s goes to 1. These results are contained in [3], a joint paper with G. Palatucci.
References:
[1] A. Di Castro, T. Kuusi, G. Palatucci: Nonlocal Harnack inequalities. J. Funct. Anal. 267 (2014), no. 6, 1807–1836.
[2] M. Kassmann: Harnack inequalities and Hölder regularity estimates for nonlocal operator revisited. SFB 11015-preprint (2011). Available at https://sfb701.math.uni-bielefeld.de/ preprints/sfb11015.pdf
[3] G. Palatucci, M. Piccinini: Nonlocal Harnack inequalities in the Heisenberg group. Calc. Var. Partial Differential Equations (2022). DOI:10.1007/s00526-022-02301-9

Federica Sani
“Asymptotics for a class of parabolic equations with critical polynomial and exponential nonlinearities“
We consider the Cauchy problem for a heat equation with initial data in H¹(R^N) and inhomogeneous nonlinearity. More precisely, we consider polynomial nonlinearities when N≥3 and exponential nonlinearities when N=2 which are critical in the energy space H¹(R^N) according to the Sobolev and the Trudinger-Moser inequality respectively. By means of energy methods, we study the asymptotic behavior of solutions with low energies, and we show that the splitting between blow-up and global existence is determined by the sign of a suitable functional. In the 2-space dimensional case, this functional is related to the corresponding Trudinger-Moser inequality. This is a joint work with Michinori Ishiwata, Bernhard Ruf, and Elide Terraneo.

Yuzhe Zhu
“Boundary hypoelliptic regularization in kinetic equations”
We will discuss the renormalization and regularization properties of the solutions to kinetic Fokker-Planck equations in bounded domains.