Colloquium
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 nspace, 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/(n1)! d^{n1} 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^{n1}.
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.
