THE MALLAT FAMILY FUND FOR RESEARCH IN MATHEMATICS
invites you to a
SPECIAL LECTURE SERIES
to be presented by
Linkoping University, Sweden and Ohio State University, Columbus
The lectures will be held in
Amado Mathematics Building
Technion - Israel Institute of Technology
Lecture I: Wednesday, 16 June 2004 at 15:30
WIENER TEST FOR HIGHER ORDER ELLIPTIC EQUATION
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear divergence and nondivergence equations with discontinuous coefficients, equations with degenerate quadratic form, quasilinear and fully nonlinear equations, as well as equations on Riemannian manifolds, graphs, groups, and metric spaces. A common feature of these equations is that all of them are of second order, and there have been no Wiener type characterizations for higher order equations so far. Indeed, the increase of the order results in the loss of the maximum principle, Harnack's inequality, barrier techniques, and level truncation arguments, which are ingredients in different proofs related to the Wiener test for the second order equations.
In the present talk I deal with an elliptic differential operator of an arbitrary even order 2m with constant real coefficients. I introduce a notion of regularity of a boundary point with respect to the Dirichlet problem which is equivalent to that given by Wiener in the case m=1. I find a necessary and sufficient condition for the regularity stated in terms of capacity which includes Wiener's result as a particular case. Some challenging open problems will be discussed as well.
Lecture II: Thursday, 17 June, 2004 at 15:30
NEW SPECTRAL CRITERIA FOR THE SCHRODINGER OPERATOR
This lecture is a survey of the conditions on the potential responsible for various spectral properties of the Schrodinger operator: positivity and strict positivity, semi-boundedness, discreteness of the spectrum, form-boundedness, finiteness and discreteness of the negative spectrum, etc.
Lecture III: Sunday, 20 June, 2004 at 15:00
ASYMPTOTIC THEORY OF HIGHER ORDER OPERATOR DIFFERENTIAL EQUATIONS
WITH NONSMOOTH NONLINEARITIES
An asymptotic theory of quasilinear operator differential equations of an arbitrary order is developed. It is shown that the main term in an asymptotic representation of solutions at infinity satisfies a finite dimensional dynamical system perturbed by a small nonlocal operator. Estimates and explicit asymptotic formulae for solutions are obtained. The results are applied to the description of boundary singularities of solutions to semilinear elliptic equations. The boundary is not necessarily smooth. The main ingredient of the proof is a spectral splitting and reduction of the original problem to the above mentioned dynamical system. This is a joint work with V. Kozlov.