Seminari in programmazione durante l'anno accademico 2025-2026

Alessandro Pigati (Università Bocconi)

Anisotropic Allen-Cahn and convergence to anisotropic area

5 dicembre 2025 alle ore 12:30: in aula 1BC45

In this talk we will introduce a PDE way to construct hypersurfaces which are critical for a generalization of area, based on an anisotropic integrand. Namely, we study energy concentration for rescalings of an anisotropic version of Allen-Cahn.

Besides a Gamma-convergence result, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn) concentrates along an integer rectifiable varifold, a weak notion of hypersurface, using stability (or finite Morse index) to compensate for the lack of monotonicity formulas.

Among the technical ingredients, we will see a generalization of Modica's bound and a diffuse version of the stability inequality for hypersurfaces.

This is joint work with Antonio De Rosa (Bocconi University).

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Cyrill Muratov (Università di Pisa)

Skyrmions in ultrathin magnetic films: an overview

19 novembre 2025 alle ore 11:30: in aula 2AB45

I will present an overview of the current results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy  and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.

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Michele Caselli  (SNS Pisa)

Saddle-type solutions to phase-transition models

28 ottobre 2025 alle ore 11:00: in aula 1BC45

In this talk, we will provide an overview of results on entire saddle-type solutions to phase transition models, as the Allen–Cahn equation in codimension one or the Ginzburg–Landau system in codimension two, highlighting similarities and differences. Then, we will present a new result, obtained in a joint work with Nicola Picenni, regarding the existence of an entire saddle solution of the complex Ginzburg–Landau system in 3D whose zero set is a union of two orthogonal, intersecting lines.

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Eugenio Bellini  (Università di Padova)

Curvature measures and the sub-Riemannian Gauss-Bonnet Theorem

8 ottobre 2025 alle ore 12:30: in aula 1BC45

 It is not uncommon for curvature to concentrate at the singularities of geometric spaces. In this talk, we show how this phenomenon occurs for surfaces immersed in 3D contact sub-Riemannian manifolds. Adopting a measure-theoretic viewpoint on the Riemannian approximation scheme, we prove that the Gaussian curvature measure of such a surface is singular and supported on its isolated characteristic points. We identify natural geometric conditions under which this behavior occurs, namely when the surface admits characteristic points of finite order of degeneracy.

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Isabeau Birindelli  (Università di Roma Sapienza)

Fully nonlinear equations in thin domains, à la Evans

25 settembre 2025 alle ore 11:30: in aula 2AB45

We will present some recent works and works in progress with Ariela Briani and Hitoshi Ishii, where we consider the asymptotic behaviors of solutions of equations in family of domains which "lose" one dimension at the limit. Starting from new proofs for old results and obtaining new insights in particular for general oblique boundary conditions. The analysis is done through the perturbed test function approach à la Evans.

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Luca Capogna  (Smith College)

The Neumann problem in metric measure spaces and  applications to non-local p-Laplacians

18 settembre 2025 alle ore 12:30: in aula 1BC50

We review the ongoing program joint with Gibara, Korte, and Shanmugalingam, aimed at extending the Caffarelli-Silvestre approach to non-local operators from the Euclidean space to general doubling metric measure spaces. An integral component of this program is the study of the well-posedness of the Neumann problem in uniform domains in metric measure spaces that satisfy a Poincare' inequality and a doubling condition.

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