Speaker
Prof.
Marie Berge Stine
(Norwegian University of Science and Technology)
Description
In the 70's Almgren noticed that for a harmonic real-valued function defined on a ball, its L^2-norm over spheres with increasing radius will have an increasing logarithmic derivative with respect to the radius of the mentioned sphere. A similar fact holds true for several other classes of functions which are solutions to PDE's, for example, Helmholtz equation, Schrödinger equation, and elliptic equations. We begin by showing how one can go from harmonic functions to solutions of the Helmholtz equation. Thereafter, we examine similar integrals over a more general class of parameterized surfaces by studying harmonic functions defined on compact subdomains of Riemannian manifolds. Some of the results are joint work with E. Malinnikova.
