Parma NEW Colloquium
The ERC StG NEW - funded colloquium series
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January 30, 2026. Susanna Terracini (Turin)
Title: Singularly perturbed elliptic systems modeling partial separation and their free boundaries
Abstract: We investigate the asymptotic behavior, as β --> ∞, of solutions to competition-diffusion system of type
where 𝜑∈𝑊^{1,∞}(Ω) satisfy the partial segregation condition
For β>1 fixed, a solutions can be obtained as a minimizer of the functional
on the set of functions in 𝐻^{1}(Ω,ℝ^{3}) with fixed traces on 𝜕Ω. We prove a priori and uniform in β Hölder bounds. In the limit, we are lead to minimize the energy
over all partially segregated states:
satisfying the given, partially segregated, boundary conditions above. We prove regularity of the free boundary up to a low-dimensional singular set.
February 25, 2026. Alessandro Iacopetti (Turin)
Title: Shape Optimization and Overdetermined Problems in Unbounded Domains
Abstract: In this talk, we present some recent results on partially overdetermined problems in unbounded regions. In particular, we focus on cones and cylinders, investigating the stability and instability of certain classes of solutions that are naturally connected to the geometry of the domain. Moreover, we discuss the existence of minimizers of the torsional energy under a volume constraint, as well as their geometric and topological properties.
These results are collected in a series of joint works with Prof. F. Pacella (University of Rome “La Sapienza”), Prof. T. Weth (University of Frankfurt), Dr. D. Gregorin (University of Urbino), and Prof. P. Caldiroli (University of Turin).
March 25, 2026. Andrea Malchiodi (SNS)
Title: The Yamabe problem
Abstract: The Yamabe problem is a generalization of the classical Uniformization problem to manifolds of dimension greater or equal to three. It was formulated in 1960, and
represents a model for geometric PDEs with lack of compactness. While it was completely solved in the 1980s thanks to the contributions of Trudinger, Aubin and Schoen, more recently it has revealed some surprising aspects. We will discuss the main results in the literature, as well some perspectives and open problems.
April 14, 2026. Riccardo Tione (Turin)
Title: Hyperbolic regularization effects for degenerate elliptic equations
Abstract: In this talk, I will address a question raised by D. De Silva and O. Savin concerning the regularity of planar solutions to div(Df(Du)) = 0 under the sole assumption that f is C^1 is strictly convex and u is Lipschitz continuous. After a brief overview of the literature on such highly degenerate equations, starting from the foundational work of De Silva and Savin, I will focus on new regularity results obtained in collaboration with X. Lamy. These show, essentially, that u is C^1 regular up to an isolated set of points provided f is fully degenerate only along C^1 curves, an extension of the previously known results that required f to be fully degenerate only at isolated points. I will explain some of the main ideas of the work, in particular the connection of this problem with Hamilton-Jacobi equations, and, if time allows, some details of the arguments.
June 18, 2026. José Mazón (Valencia)
Title: The problem of infinite propagation speed in diffusion equations
Abstract: One of the most widely used mathematical tools in modeling is the class of diffusion equations, which are present not only in physical, chemical, and biological models, but also in virtually every scientific field. For example, the now famous Black–Scholes equation for European call options is nothing but a diffusion equation in which what diffuses are prices. Although it is well known that linear diffusion models based on Fick’s law
(just as the heat equation based on Fourier’s law) lead to the physical contradiction of infinite propagation speed, they are still the most commonly used models whenever some diffusion process is involved. In certain specific problems, the models obtained using these linear equations provide a good approximation to reality, despite the aforementioned physical contradiction, because the solutions, although positive, are very small outside a compact set. However, in many biological models, such as morphogen transport, the infinite propagation speed completely invalidates the model.
In this talk, after presenting a historical overview of the different diffusion models, we introduce some of the results we have obtained concerning a diffusion equation proposed by Ph. Rosenau, and independently by Y. Brenier. This equation, referred to by Y. Brenier as the relativistic heat equation, has finite propagation speed and, more interestingly, the maximum propagation speed is a parameter of the equation itself. Therefore, it can be predetermined according to the nature of the problem under study.
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