Some properties of a harmonic function in the neighborhood of an isolated singularity

[摘要] Let R¯ be a closed and bounded region of the plane, let P be an interior point of R¯. Let u1(r,theta) be a function single valued and harmonic in R = R¯ - {P} and having an isolated singularity at the point P. For purposes of the work to follow there will be no loss in generality to take P as the origin (r = o) and R as the region R : o < r ≤ 1, o ⩽ theta ≤ 2pi. If the function u1(r,theta) is normalized in R- i.e. if we subtract from u1(r,theta) a function which is single valued and harmonic in R¯ and assumes on r = 1 the values u1(1,theta)- we obtain a function u(r,theta) satisfying the conditions: (a) u(r,theta) is single valued and harmonic in R. (b) u(1,theta) ≡ o in theta. (c) u(r,theta) has an isolated singularity at r = o. Consider the sets of points of R defined as follows: (1) D + the set of points of R for which u(r,theta) > o. (2) D - the set of points of R for which u(r,theta) < o. (3) D o the set of points of R for which u(r,theta) = o. The problems to be considered in the following work are (1) the characterization of the point sets D+, D-, and Do and (2) the behavior of u(r,theta) in these point sets in a neighborhood of r = o. We shall find that the point sets D+, D- and Do can be resolved into components, i.e. connected subsets, and with at most one exception every component of D+ and D - is a simply connected domain.

[发布机构] Rice University