3:30 p.m., Friday

Math 100

Professor Peter Li

Department of Mathematics

University of California, Irvine

Sharp Asymptotic Bounds on the Dimensions of Harmonic Functions

In Euclidean n-space, the set of homogeneous harmonic polynomials spans the space of all harmonic functions that grows polynomially. In particular, if we denote H_d(R^n) to be the space of harmonic functions that grows at most polynomially of degree d, then by counting homogeneous harmonic polynomials we obtain that dim H_d(R^n) is tending to 2/(n-1)! d^{n-1} as d goes to infinity. It turns out that for a large class of complete manifolds M one can show that the space, H_d(M), of polynomial growth harmonic functions of degree at most d, must satisfy

dim H_d(M) \leq C d^{n-1}.

An interesting question is to determine what is the best possible value of the constant C and what is its geometric significance. The purpose of this talk is to give some historical aspect of this problem, and to provide answers to some special cases. A parallel theory with similar type questions is also valid for uniformly elliptic operators of divergence form with measurable coefficients.

Refreshments will be served in Math Annex Room 1115, 3:15 p.m.

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