Eigenfunction and harmonic function estimates in domains with horns and cusps
Michael D. Cranston, Yi Li · Communications in Partial Differential Equations · 1997
this paper we shall obtain pointwise bounds for positive harmonic functions vanishing 1 on the lateral side of horn-shaped domains in R d as well as for Dirichlet eigenfunctions in these domains. Our most general horn-shaped domain is as follows. Let R : [0, #) ## U(d), the unitary group, S : [0, #) ## R d-1 , f : [0, #) ## (0, #) and suppose all are continuous (more conditions will be added in due course) and take# to be a bounded domain in R d-1 . Then our domains are of the form D = {(x, y) : x > 0, R(x)(y - S(x)) # f(x) ## } . Throughout this paper we will always assume that lim x## f(x) = 0 and ## # C 2,# , for some # > 0. Suppose w 1 is the first Dirichlet eigenfunction of the Laplacian for# and