In this talk, we will consider the p-Dirichlet energy of maps with values in a closed manifold. We will discuss a Gamma-convergence result in the limit as the exponent p approaches a certain critical value k, which is defined in terms of the topology of the target manifold. This particular limit is associated with the emergence of topological singularities of codimension k, which can be...
The Plateau problem concerns the surfaces of least m-dimensional area spanning a given (m-1)-dimensional boundary. To guarantee existence of minimizers and desirable compactness properties for sequences of surfaces, one must consider a weak notion of surface, which allows area-minimizing "surfaces” to have singularities. Two particularly natural frameworks for this problem in arbitrary...
We present some aspects of the theory of nonlocal minimal surfaces, with special attention to regularity, sheeting, maximum principle, and stickiness. We also link this theory to the study of long-range phase transition and to the analysis of its symmetry properties. Similarities and differences with the classical cases will be outlined.
Suppose that two nonlocal minimal surfaces are included one into the other and touch at a point. Then, they must coincide. But this is perhaps less obvious than what it seems at first glance.
We study the Gamma-convergence of Ambrosio-Tortorelli-type functionals, for maps u defined on an open bounded set Ω ⊂ R^n and taking values in the unit circle S^1 ⊂ R^2. Depending on the domain of the functional, two different Gamma-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map u whose measure of the jump set is...
We will discuss recent progress in the regularity for stable solutions to the Allen-Cahn equation, with a focus on the connections with minimal surface theory and two influential conjectures of De Giorgi and Yau. Part of this talk is based on recent joint work with Joaquim Serra which classifies all stable solutions to Allen-Cahn with bounded energy density in four dimensions.
