Infinite dimension of solutions of the Dirichlet problem

Abstract It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.


Introduction
By the well-known Lindelöf maximum principle, see e.g. Lemma 1.1 in [3], it follows the uniqueness theorem for the Dirichlet problem in the class of bounded harmonic functions u on the unit disk D D fz 2 C W jzj < 1g. In general there is no uniqueness theorem in the Dirichlet problem for the Laplace equation even under zero boundary data. In comparison with [7], here we give more elementary examples and constructions of solutions.
Example 1.1. The first natural example is given by the formula (1) withˆ.t / D '.t =2 / where ' W OE0; 1 ! OE0; 1 is the well-known Cantor function, see e.g. [1] and further references therein. Example 1.2. However, the simplest example of such a kind is given by nondecreasing step-like dataˆ# 0 with values 0 and 2 and with the jump at # 0 2 .0; 2 /: We see that u.z/ ! 0 as z ! e i ‚ for all ‚ 2 .0; 2 / except ‚ D # 0 . Note that the function u is harmonic in the unit disk D because where the function w D g.z/ D g 0 .z/W D . 0 Cz/=. 0 z/ is analytic (conformal) in D and maps D onto half-plane Re w > 0, g.0/ D 1, g. 0 / D 1.

The main result
The formula (2) gives a continual set of such examples. Furthermore, one can prove the following result.
Theorem 2.1. The space of all harmonic functions in D with nontangential limit 0 at every point of @D except a countable collection of points in @D has the infinite dimension.
Proof. Indeed, let us consider the sequence of functions of the form (3): where # n D .2 1 C : : : C 2 n /; n D 1; 2; : : : and denote by H 1 the class of all series u D P n u n whose sequences of coefficients D f n g belong to the space l 1 with the norm k k D for all points z D re i # belonging to a sector j# ‚j < c.1 r/ and for all r which are close enough to 1 where C < 1 does not depend on n D 1; 2; : : :. Thus, in any sector j# ‚j < c.1 r/. Now, let us show that u n , n D 1; 2; : : :, form a basis in the space H 1 with the locally uniform convergence in D which is metrizable.
Indeed, firstly, u D 1 P nD1 n u n ¤ 0 if ¤ 0. Really, let us assume that n ¤ 0 for some n D 1; 2; : : :. Then u ¤ 0 because u.z/ ! 1 as z D re i # n ! e i # n . The latter follows because u n .re i # n / D 1 C r 1 r ! 1 as r ! 1; and by the previous item j Q u.re i # n /j Ä C k k .1 r/ ! 0 as r ! 1; where Q u D u n u n . in every disk D.r/ D fz 2 C W jzj Ä rg, r < 1. Indeed, the existence at least one such a harmonic function u follows from the known Gehring theorem in [4]. Combining this fact with Theorem 2.1, we obtain the conclusion of Corollary 3.1.

Corollaries and final remarks
Remark 3.2. In view of Lemma 3.1 in [2], one can similarly prove the more refined result on harmonic functions than in Corollary 3.1 with respect to logarithmic capacity instead of the measure of the length on @D.
Moreover, the statements on the infinite dimension of the space of solutions can be extended to the Riemann-Hilbert problem because the latter is reduced to the corresponding two Dirichlet problems as in papers [2] and [7].
Note also that harmonic functions u found in Theorem 2.1 and Corollary 3.1 themselves cannot be represented in the form of the Poisson integral with any integrable functionˆW OE0; 2 ! R because such integral would have nontangential limitsˆa.e., see e.g. Corollary IX.1.1 in [5]. Consequently, u do not belong to the classes h p for any p > 1, see e.g. Theorem IX.2.3 in [5].
However, the functions u 2 H 1 in the proof of Theorem 2.1 have the representation as the Poisson-Stiltjes integral (1) withˆD P nˆ# n whereˆ# n W OE0; 2 ! R are nondecreasing step-like functions with values 0 and 2 with jumps at the points # n , n D 1; 2; : : :. Thus,ˆis of bounded variation and hence H 1 h 1 , see e.g. Theorem IX.2.2 in [5].