Thomas Bieske, University of Michigan

Lipschitz Extensions on Grushin-type spaces.

Abstract: In this talk, we examine viscosity solutions and absolute minimizers in Grushin-type spaces, which are sub-Riemannian spaces without a group structure.  We show that with a regularity condition, absolute minimizers are viscosity infinite harmonic.

Marian Bocea, Carnegie Mellon University

The structure of minimizing sequences for 3D-2D dimensional reduction problems.

Abstract: It is shown that the Dirichlet problem on arbitrarily large cylinders with fixed affine lateral boundary conditions admits p-equi-integrable minimizing sequences energetically preferring thinner and thinner reference domains. The proof uses L^p estimates for maximal functions, fine truncation results for Sobolev maps, and De Giorgi's slicing method. Joint work with Irene Fonseca (Carnegie Mellon University).

Filippo Gazzola, Università del Piemonte
Web functions and their minimizing properties.

Abstract: Let Ω be a convex bounded set in Rⁿ. We call web function in Ω any function which depends only on the distance from the boundary of Ω. We consider the problem of minimizing a class of (possibly nonconvex) functionals and we approximate their infimum in the usual Sobolev spaces with their minimum in the space of web functions. The latter exists without any convexity assumption. We show that this approximation is good for several meaningful models, such as the torsion problem and its generalizations.

Tiziana Giorgi, New Mexico State University
Superconductors surrounded by normal materials.

Abstract: We study questions related to existence in suitable weighted Sobolev spaces, and to properties of minimizers of a generalized energy functional, which models a bounded superconductor surrounded by a normal material.The model in consideration is of interest as the effects of superconducting electron pairs diffusing into the normal region are here represented.
 

Bo Guan, University of Tennessee
Monge-Ampere equations with infinite boundary value.

Abstract: I will report on joint work with H-Y. Jian on the problem. In this work we give growth conditions that are nearly optimal for the existence of solutions.

David Hoff, Indiana University

Compressible flow in regions with rough boundaries.

Abstract: I'll describe the construction of solutions of the Navier-Stokes equations of compressible flow with the "rough-boundary" condition proposed by Navier: on the boundary, the velocity should be proportional to the tangential stress. I'll explain what this means, and how this assumption provides exactly the right ingredient for extending the theory of solutions in the whole space to solutions in domains with boundary.

Xiaosheng Li, University of Illinois, Urbana – Champaign

Schottky quasiconformal groups in metric spaces.

Abstract: In this talk, we discuss quasiconformal mappings in metric spaces. Specifically, we consider Schottky quasiconformal groups. The main result is that the limit set of any Schottky quasiconformal group in certain metric space is uniformly perfect.

Yoshihiro Tonegawa, Hokkaido University

On some results on the two-phase singular perturbation problem.

Abstract: I will describe some known results on the singular perturbation problem initially studied by Sternberg, Modica, and others and some recent results on the non-minimal critical points as well as stable critical points via the method of geometric measure theory.

Changyou Wang, University of Kentucky
Regularity of biharmonic maps.

Abstract:  In this talk, I plan to sketch the main ideas to prove the smoothness of both intrinisc and extrinsic biharmonic maps into general Riemannian manifolds in dimension four and partial regularity in dimensions five or beyond.