Mark Agranovsky
Bar Ilan University, Israel

Global geometry of stationary sets

The subject of the talk is characterizing of stationary sets -- i.e. sets where a non-trivial solution is zero at any moment of time -- for evolution (heat and wave) equations in R^n or in bounded domains. For time-harmonic (i.e., single mode) solutions stationary sets coincide with nodal (zero) sets of eigenfunctions of the Laplace operator.

Understanding of the geometry of nodal sets is a difficult problem, and mostly information about only the local structure of nodal sets is available. Nevertheless, when the solution has sufficiently rich spectrum (i.e., involves infinitely many time harmonic modes) and correspondinlgly the stationary sets are intersections of infinite families of nodal sets of the eigenfunctions in the spectrum, the global geometric structure of stationary sets can be well understood and in certain situations completely described. That is the case, for instance, when the initial velocity vanishes near the boundary of a bounded domain, or in the case of the whole space has compact support or decays sufficiently fast at infinity.

Results of recent work on the subject will be presented.