February 28  March 2, 2000
David R. Adams (Lexington)
Abstract: We consider the classical obstacle problem for the biharmonic operator on a bounded smooth domain in Euclidean nspace (with zero Dirichlet boundary conditions) and for a smooth obstacle function given on the domain which is negative at the boundary of the domain. The problem addressed is that of determining conditions that insure that the solution of the biharmonic obstacle problem for a sequence of obstacles converges to the solution of the biharmonic obstacle problem for the limit obstacle. It is well known that if the convergence of the sequence of obstacles is too weak, then the corresponding solutions will not converge to the desired limit problem. Such questions have been treated by Gamma convergence methods. However, it is not known when that theory yields results of the type outlined here. The methods discussed in this talk will produce the desired limit. They make use of certain maximal functions from Harmonic Analysis. Several interesting examples result from this approach. Ioannis Athanasopoulos (Herakleion)
Abstract: In joint work with L.A.Caffarelli and S.Salsa we consider parabolic free boundary problems where the velocity of the free boundary depends also on its curvature. In contrast with the corresponding ones where the curvature is absent the free boundary does not exhibit hyperbolic phenomena. In fact we prove that Lipschitz free boundaries are smooth. Luis A. Caffarelli (Austin)
Donatella Danielli (Baltimore)
Abstract: In this talk I will discuss the uniform properties, and the limit as , of solutions of the following parabolic singular perturbation problem: Nicola Garofalo (Baltimore)
Qing Han (Notre Dame)
Abstract: We will discuss the geometric structure and geometric measure of singular sets of solutions to general elliptic equations. We will show that such sets are countably rectifiable under very weak assumptions on coefficients. We will also show the measure estimates on these sets if we have the smoothness of the coefficients. The measure estimates are obtained in terms of frequency of solutions. A key ingredient is the monotonicity formula by GarofaloLin. The method is from algebraic geometry and geometric measure theory. Last we will indicate how to improve such estimates for analytic solutions, especially harmonic functions Håkan Hedenmalm (Lund)
Abstract: The exponential mapping of differential geometry is a local homeomorphism from the tangent plane to the manifold, defined so that geodesics (i. e., straight lines) through the origin in the tangent plane are mapped to geodesics through the given point, so that distances are preserved. A classical theorem of Hadamard (1898) states that in two dimensions, the exponential mapping is a global homeomorphism onto the surface provided the surface is simply connected and has negative Gaussian curvature everywhere. We extends Hadamard's theorem to the context of HeleShaw flow (the classical theorem concerns metric flow), which corresponds to a Newtonian fluid instead of a photonic gas. New problems arise, such as that of ergodicity of Wrapped HeleShaw flow on compact negatively curved surfaces. David Jerison (Cambridge)
Abstract: We will discuss joint work with L. A. Caffarelli and C. E. Kenig on existence and regularity of elliptic free boundary problems. These problems are motivated by the example of the PrandtlBatchelor problem in two variables. The novelty is that the inhomogeneous term can be discontinuous across the free boundary. Lavi Karp (Haifa)
Abstract: Suppose is a bounded domain in with real analytic boundary , and suppose f is a real analytic function in a neighborhood of . Then, as a consequence of Cauchy Kovalewskii's theorem, the Newtonian potential of the density f over (which is harmonic out of ), has a harmonic continuation into . Bernd Kawohl (Köln)
Abstract: The wellknown FaberKrahn inequality states that among all clamped membranes of given area the circular one has minimal fundamental frequency. In my lecture I shall recall different methods of proof for it and report on related results for plate eigenvalues. These lead to new questions on the regularity of sets which minimize eigenvalues. Claudia Lederman (Buenos Aires)
Abstract: We consider the following two phase free boundary problem: find a function u(x,t), defined in , satisfying that Peter A. Markowich (Vienna)
Abstract: Mean field equations describe thermal equilibrium states of interacting species of particles in plasmas. Typically, these equations take the form of a semilinear elliptic equation with a nonlocal L1nonlinearity. We show that mean field equations arise as large time limits of (singular) diffusion equations with Poisson coupling. The main tool in this proof are generalized Sobolev inequalities and the main difficulty in the proof lies in the limited regularity due to the occurance of free boundaries in the porous medium case. Also, the singular perturbation limit is analysed  leading, in certain parameter regimes, to a double obstacle problem. Here, the free boundaries are of particular physical importance since they separate vacuum from nonvacuum regions. The singular perturbation analysis is based on the dual variational formulation with a functional which is not necessarily coercive (in the fast diffusion case).Particular emphasis is given to the convergence (and geometry) of the free boundary in the singular perturbation limit (porous medium case). The talk is based on a joint papers with L. Caffarelli, J. Dolbeault, C. Schmeiser and A. Unterreiter. Jorge Salazar (Lisbon)
Abstract: We study some properties of viscosity solutions of fully nonlinear elliptic equations of the form Sandro Salsa (Milan)
Abstract: We describe some recent results obtained in a joint work with Athanasopoulos and Caffarelli concerning a class of free boundary problems and their application to the uniqueness in a classical inverse problem of conductivity. The regularity of the free boundary in analyzed in details under minimal conditions. Nina Ural'tseva (St. Petersburg)
Abstract: The behaviour of free boundaries near the prescribed ones is studied for some classes of free boundary problems. In the case of zero obstacle problem it is proved that the free boundary touches tangentially the given boundary provided the latter is smooth and the boundary values of the solution vanish in a neighbourhood of a contact point. For more general free boundary problem, without sign restrictions on a solution, the same behaviour of the free boundary near contact points is established under certain conditions. Georg S. Weiss (Tokyo)
Abstract: In dimension we obtain regularity of the free boundary of nonnegative solutions of the heat equation with strong absorption

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