Anisotropic Allen-Cahn and convergence to anisotropic area

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).

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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