PDE Geometric Analysis seminar

PDE and Geometric Analysis Seminar - Fall 2010

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Seminar Schedule

date	speaker	title	host(s)
Sept 13	Fausto Ferrari (Bologna)	Semilinear PDEs and some symmetry properties of stable solutions	Misha
Sept 27	Arshak Petrosyan (Purdue)	Nonuniqueness in a free boundary problem from combustion	Misha
Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. Special day, time & room.	Changyou Wang (U. of Kentucky)	Phase transition for higher dimensional wells	Misha
Oct 11	Philippe LeFloch (Paris VI)	Kinetic relations for undercompressive shock waves and propagating phase boundaries	Misha
Oct 29 Friday 2:30pm, Room: B115 Van Vleck. Special day, time & room.	Irina Mitrea (IMA)	Boundary Value Problems for Higher Order Differential Operators	WiMaW
Nov 1	Panagiota Daskalopoulos (Columbia U)	Ancient solutions to geometric flows	Misha
Nov 8	Maria Gualdani (UT Austin)	TBA	Misha
Date TBA	Mikhail Feldman (UW Madison)	TBA	Local speaker
Date TBA	Sigurd Angenent (UW Madison)	TBA	Local speaker

Abstracts

Fausto Ferrari (Bologna)

Semilinear PDEs and some symmetry properties of stable solutions

I will deal with stable solutions of semilinear elliptic PDE's and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.

Arshak Petrosyan (Purdue)

Nonuniqueness in a free boundary problem from combustion

We consider a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. V ́azquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion.

This is a joint work with Aaron Yip.

Changyou Wang (U. of Kentucky)

Phase transition for higher dimensional wells

For a potential function [math]\displaystyle{ F }[/math] that has two global minimum sets consisting of two compact connected Riemannian submanifolds in [math]\displaystyle{ \mathbb{R}^k }[/math], we consider the singular perturbation problem:

Minimizing [math]\displaystyle{ \int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right) }[/math] under given Dirichlet boundary data.

I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic, as the parameter [math]\displaystyle{ \epsilon }[/math] tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.

Philippe LeFloch (Paris VI)

Kinetic relations for undercompressive shock waves and propagating phase boundaries

I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics. In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.

Irina Mitrea

Boundary Value Problems for Higher Order Differential Operators

As is well known, many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator L in a domain D.

When L is a differential operator of second order a variety of tools are available for dealing with such problems including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. The situation when the differential operator has higher order (as is the case for instance with anisotropic plate bending when one deals with fourth order) stands in sharp contrast with this as only fewer options could be successfully implemented. Alberto Calderon, one of the founders of the modern theory of Singular Integral Operators, has advocated in the seventies the use of layer potentials for the treatment of higher order elliptic boundary value problems. While the layer potential method has proved to be tremendously successful in the treatment of second order problems, this approach is insufficiently developed to deal with the intricacies of the theory of higher order operators. In fact, it is largely absent from the literature dealing with such problems.

In this talk I will discuss recent progress in developing a multiple layer potential approach for the treatment of boundary value problems associated with higher order elliptic differential operators. This is done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory.

Panagiota Daskalopoulos (Columbia U)

Ancient solutions to geometric flows

We will discuss the clasification of ancient solutions to nonlinear geometric flows. It is well known that ancient solutions appear as blow up limits at a finite time singularity of the flow. Special emphasis will be given to the 2-dimensional Ricci flow. In this case we will show that ancient compact solution is either the Einstein (trivial) or one of the King-Rosenau solutions.

Maria Gualdani (UT Austin)

TBA