20 may 2026 [open in google calendar]
Agenda: Get-together (30 min), presentation Alessandro Pigati (60 min), questions and discussions (30 min).
Abstract.
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.
After reviewing known existence and regularity results for minimizers and min-max solutions, as well as the Allen-Cahn approximation in the isotropic case, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn functional) 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).