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 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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