Speakers

I will present a result obtain in collaboration with Hitoshi Ishii and Ariela Briani, that studies the asymptotic behaviors of solutions of fully nonlinear equations in thin domains with oblique boundary conditions with a test function approach à la Evans. The limit equation contains new terms in the second, first and zeroth order terms.

The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. This holds true both in the degenerate range $m>1$ and in the singular range $0<m<1$. The critical and sub-critical singular case, i.e. when $0<m\le \frac{(N-2)_+}{N+2}$, presents further difficulties, which have been recently settled. I will discuss the problem in its generality, focusing in particular on the latest results. The critical and subcritical case is a joint work with V. B\"ogelein, F. Duzaar and. N. Liao (University of Salzburg, Austria), but previous contributions are also due to S. Schwarzacher, R. Korte, C. Scheven.

We consider viscosity solutions to a two-phase free boundary problem for a nonlinear elliptic PDE with non-zero right hand side. We obtain regularity results for solutions and their free boundaries. The operator under consideration is a model case in the class of partial differential equations with non-standard growth. This type of operators have been used in the modelling of non-Newtonian fluids, such as electrorheological or thermorheological fluids, also in non-linear elasticity and image reconstruction. The fact that we are dealing with nonlinear degenerate/singular equations with non-zero right hand side leads to challenging difficulties that will be addressed in this talk. This is joint work with Fausto Ferrari (University of Bologna, Italy).

In this talk, we revisit some classical Liouville-type results in half spaces for uniformly elliptic equations. We extend these results to the nonuniformly elliptic equations known as false mean curvature equations. The regularity theory of locally Lipschitz viscosity solutions will also be addressed. ‎

We discuss the construction of some new family of homogenous solutions for the thin obstacle problem in R^3, with frequencies different from the standard ones coming from R^2. This is a joint work with Federico Franceschini.

We investigate the asymptotic behavior, as $\beta \to +\infty$, of solutions to competition-diffusion system of type \[ \begin{cases} \Delta u_{i,\beta} = \beta u_{i,\beta} \prod_{j \neq i} u_{j,\beta}^2 & \text{in }\Omega,\\ u_{i,\beta} = \varphi_i \ge 0& \text{on }\partial \Omega, \end{cases} \quad i=1,2,3, \] where $\varphi_i \in W^{1,\infty}(\Omega)$ satisfy the \emph{partial segregation condition} \[ \varphi_1\,\varphi_2\,\varphi_3 \equiv 0 \quad \text{in $\overline{\Omega}$}. \] For $\beta>1$ fixed, a solutions can be obtained as a minimizer of the functional \[ J_\beta({\bf u},\Omega):= \int_{\Omega} \big( \sum_{i=1}^3 |\nabla u_i|^2 + \beta \prod_{j=1}^3 u_j^2\big)\,dx \] on the set of functions in $H^1(\Omega,\R^3)$ with fixed traces on $\partial \Omega$. We prove \emph{a priori} and \emph{uniform in $\beta$} H\"older bounds. In the limit, we are lead to minimize the energy \[ J{\bf u},\Omega):= \int_{\Omega} \sum_{i=1}^3 |\nabla u_i|^2 \,dx \] over all partially segregated states: \[ u_1\,u_2\,u_3 \equiv 0 \quad \text{in $\overline{\Omega}$} \] satisfying the given, partially segregated, boundary conditions above. We prove regularity of the free boundary up to a low-dimensional singular set.

When analyzing the survival threshold for a species in population dynamics, one is led to consider the principal eigenvalue of some indefinite weighted problems in a bounded domain. The minimization of such eigenvalue, associated with either Dirichlet or Neumann boundary conditions, translates into a shape optimization problem. We perform the analysis of the singular limit of this problem, in case of arbitrarily small favorable region. We show that, in this regime, the favorable region is connected and it concentrates at points depending on the boundary conditions. Moreover, we investigate the interplay between the location of the favorable region and its shape. Joint works with Lorenzo Ferreri, Dario Mazzoleni and Benedetta Pellacci.

The Stefan problem describes the phenomenon of freezing and melting of a material with a solid-liquid interphase, such as ice and water. It can be formulated as a parabolic free boundary problem where the interface between the solid and the liquid regions evolves over time and is an unknown of the problem. After introducing the problem and some related literature, the seminar will focus on presenting some recent techniques to obtain regularity results starting from an initial suitable flatness assumption for the one-phase Stefan problem with the presence of a right-hand side. This talk will be based on a recent collaboration with F.Ferrari, N.Forcillo and D.Jesus.

Transmission problems describe physical phenomena where the behavior of a system changes across a fixed interface, resulting in different PDEs on each side. These PDEs are coupled through a transmission condition, typically of the Neumann type. In this talk, we will discuss some recent results concerning transmission problems governed by fully nonlinear operators which degenerate near the transmission interface. We obtain Holder differentiability of solutions up to the interface, with optimal exponent, which depends pointwise on the rate of degeneracy.

The regularity of minimizers for the classical one-phase Bernoulli functional has been extensively studied following the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers has also been investigated. We present some results concerning the regularity of almost minimizers for a one-phase Bernoulli-type functional within Carnot groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. We outline the main challenges and techniques employed to establish local Lipschitz regularity for almost minimizers, emphasizing the role of intrinsic gradient estimates. Additionally, we discuss certain generalizations to the nonlinear framework. Some of the results presented stem from joint work with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).