PDE Geometric Analysis seminar

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.

Previous PDE/GA seminars

Seminar Schedule Fall 2011

date	speaker	title	host(s)
Oct 3	Takis Souganidis (Chicago)	Stochastic homogenization of the G-equation	Armstrong
Oct 10	Scott Armstrong (UW-Madison)	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)	The pentagram map and generalized friezes of Coxeter	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)	TBA	Armstrong
Dec 5	Charles Smart (MIT)	TBA	Armstrong

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.

Scott Armstrong (UW-Madison)

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.

Adrian Tudorascu (West Virginia University)

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)

To be posted.

Charles Smart (MIT)

To be posted.