Construction of Planar Harmonic Functions
نویسندگان
چکیده
منابع مشابه
Construction of Planar Harmonic Functions
Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk can be written in the form f = h+ g, where h and g are analytic in the open unit disk. The functions h and g are called the analytic and coanalytic parts of f , respectively. In this paper, we construct certain planar harmonic maps either by varying the coanalytic parts of harmonic functions that are...
متن کاملValence of Complex-valued Planar Harmonic Functions
The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ C. Let S denote the critical set of f and C(f) the global cluster set of f . We show that f(S)∪C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f(S) ∪ C(f) has empty interior. We also...
متن کاملC-boundary regularity of planar infinity harmonic functions
We prove that if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then any viscosity solution u ∈ C(Ω) of the infinity Laplacian equation (1.1) is C(Ω). The interior C and C-regularity of u in dimension two has been proved by Savin [20] and Evans-Savin [15] respectively. We also show that for any n ≥ 3, if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then the solution u of equ...
متن کاملA lower estimate of harmonic functions
We shall give a lower estimate of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Ya. Levin in a half plane.
متن کاملGeodesics and Bounded Harmonic Functions on Infinite Planar Graphs
It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitelyintersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph. Let g be an infinite, simple, connected...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: International Journal of Mathematics and Mathematical Sciences
سال: 2007
ISSN: 0161-1712,1687-0425
DOI: 10.1155/2007/70192