PDE and Geometric Analysis Seminar

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

Seminar Schedule Fall 2011

date speaker title host(s)
Oct 3 Takis Souganidis (Chicago)
Stochastic homogenization of the G-equation

Armstrong
Partial regularity for fully nonlinear elliptic equations

Local speaker
Oct 17 Russell Schwab (Carnegie Mellon)
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)

Armstrong
October 24 ( with Geometry/Topology seminar) Valentin Ovsienko (University of Lyon) Marí Beffa
Oct 31 Adrian Tudorascu (West Virginia University)
Weak Lagrangian solutions for the Semi-Geostrophic system in physical space

Feldman
Nov 7 James Nolen (Duke) Armstrong
Nov 21 (Joint with Analysis seminar) Betsy Stovall (UCLA) Seeger
Dec 5 Charles Smart (MIT)
Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian

Armstrong

Seminar Schedule Spring 2012

date speaker title host(s)
Feb 6 Yao Yao (UCLA)
TBA

March 19 Nestor Guillen (UCLA)
TBA

Kiselev
April 16 Jiahong Wu (Oklahoma)
TBA

Kiselev

Abstracts

Takis Souganidis (Chicago)

Stochastic homogenization of the G-equation

The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.

Partial regularity for fully nonlinear elliptic equations

I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.

Russell Schwab (Carnegie Mellon)

On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)

Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])

Valentin Ovsienko (University of Lyon)

The pentagram map and generalized friezes of Coxeter

The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map. In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.

Weak Lagrangian solutions for the Semi-Geostrophic system in physical space

Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.

James Nolen (Duke)

Normal approximation for a random elliptic PDE

I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.

Betsy Stovall (UCLA)

We will discuss recent work concerning the cubic Klein--Gordon equation u_{tt} - \Delta u + u \pm u^3 = 0 in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain questions arising in harmonic analysis.

This is joint work with Rowan Killip and Monica Visan.

Charles Smart (MIT)

Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian

A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.