On radial limit functions for entire solutions of second order elliptic equations in $\mathbf{R}^2$

Abstract

Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R 2 , we consider entire solutions of the equation Lu = 0 for whichexists for all ϕ ∈ [0, 2π) as a finite limit in C. We characterize the possible "radial limit functions" U . This is an analog of the work of A. Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Условия $C^m$-приближаемости функций решениями эллиптических уравнений

“…Этот метод, как правило, дополняется функционально-аналитическим подходом (например, методом спектрального синтеза [42]) и/или теорией сингулярных интегралов [68]. К упомянутым ра-нее ссылкам [26] и [7] мы добавим работы [112], [113] и их приложения к во-просам граничных и асимптотических значений [114] …”

Radial limits of harmonic functions on the unit disc

“…A classical result of Alice Roth [8] (or see Chapter IV, §5 of Gaier [4]) characterizes those functions on T that can be expressed as lim r→∞ f (re iθ ) for some entire function f . She showed that these are precisely the Baire-one functions on T (that is, pointwise limits of sequences from C(T)) that are constant on each (connected) component of some relatively open dense subset of T. More recently, Boivin and Paramonov [3] established (in particular) an analogue of Roth's result for harmonic functions on the plane: in this case the radial limit functions are characterized as those Baire-one functions on T that are first-degree polynomials of θ on each component of some relatively open dense subset of T. The purpose of this note is to solve the corresponding problem for radial boundary limits of harmonic functions on the unit disc. We will say that a function f : T → R is asymptotically mean-valued at e iθ if Proof.…”