Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 15-17, 2011
All talks in Vincent Hall 16
Lecture 1 - Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains.
Abstract 1: These talks discuss results on layer potentials for elliptic boundary problems, with emphasis on the Dirichlet problem for the Laplace operator. They start by reviewing results for domains with moderately smooth boundary, then for Lipschitz domains, and proceed to discuss results obtained in joint work of the speaker with S. Hofmann and M. Mitrea, for a class of domains we call regular Semmes-Kenig-Toro (SKT) domains, often called chord-arc domains with vanishing constant, and for delta-regular SKT domains, often called chord-arc domains with small constant. Lecture notes pertaining to the talk can be found on the speaker's web page.
Lecture 1 - Long-time behavior of solutions of Hamilton-Jacobi equations with Neumann type boundary conditions
Abstract 1: We discuss the long-time behavior of solutions of the convex Hamilton-Jacobi equation ut + H(x,Du) = 0 in a bounded domain Ω of Rn with the Neumann type boundary condition Dγu = g, where γ is a vector field on the boundary ∂Ω pointing a direction oblique to ∂Ω. We explain a convergence result to asymptotic solutions together with some related results concerning is the stationary problem associated with ut +H(x,Du) = 0 is: H(x,Dv) = c in Ω and Dγv = 0 on ∂Ω, where the pair, v ∈ C(Ω ̄) and c ∈ R, is the unknown.
Lecture 2 - Stochastic perturbations of Hamiltonian flows: a PDE approach
Abstract 2: We present a pde approach to the study of averaging principles for small stochastic perturbations of Hamiltonian flows in 2D, which is based on a recent joint work with P. E. Souganidis of the University of Chicago. Freidlin and Wentzel initiated the study of such problems and there have been an extensive study in this field in the last several years. Asymptotically the slow (averaged) motion has 1D character and takes place on a graph, and the question is to identify the limit motion in terms of pde problems. Our approach is based on pde techniques and applies to general degenerate elliptic operators while previous work has relied on the probabilistic techniques.
Lecture 1 - Recent progress on discrete restriction
Lecture 1 - Inelastic interaction of solitons for the quartic gKdV equation
Abstract 1: We present two recent works in collaboration with Frank Merle concerning the interaction of two solitons for the (nonintegrable) quartic (gKdV) equation. In two specific asymptotic cases (almost equal speeds / very different speeds), we can describe the collision in details. In particular, we prove that at the main order, the two solitons are preserved by the interaction as in the integrable case. However, unlike in the integrable case, we prove that the collision is inelastic.
Y. Martel et F. Merle, Description of two soliton collision for the quartic gKdV equation, to appear in Annals of Math.
Y. Martel et F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, to appear in Inventiones Mathematicae.
Lecture 1 - Optimal multidimensional pricing facing informational asymmetry
Abstract 1:The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE.
Lecture 1 - Partitions and strongly competiting systems
Abstract 1: We deal with the free boundary problem associated with optimal partitions related with linear and nonlinear eigenvalues. We are concerned with extremality conditions and the regularity of the interfaces. These properties are then linked with extremality conditions of the nodal set of eigenfunctions and the number of their nodal components.