On the Solution of the Dirichlet Problem with Rational Holomorphic ...

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446 P. Ebenfelt and M. Viscardi CMFT 4 ≤ n ≤ 270 (although, presumably, this holds also for all even n ≥ 4). In other papers (e.g. [2, 3, 8]), the analogous question with polynomials replacing rational functions was investigated on annuli, ellipsoids, and other quadratic surfaces. We mention in particular the paper [13] in which it was shown that the solutions to the Dirichlet problem (in any number of dimensions n ≥ 2) with polynomial data are always polynomial if and only if Ω is an ellipsoid. This resolves a conjecture by Khavinson and Shapiro [12]. It is natural to extend these studies by considering the question in more general domains. In this paper, we let Ω be any simply connected bounded domain in C and take the boundary data h(z, ¯z) to be a rational function of z without poles on ∂Ω. Note, in particular, that the boundary data is holomorphic in a neighborhood of ∂Ω. The case where h(z, ¯z) is allowed to be a rational function of both z and ¯z is quite different, and is the subject of the forthcoming paper [7]. We shall be concerned with characterizing those Ω for which the solution to the Dirichlet problem (1) is rational whenever the boundary data h(z, ¯z) is a rational function of z with no poles on ∂Ω. Clearly, if this happens, then the boundary of Ω is contained in a real-analytic (indeed, real-algebraic) variety. We shall here further assume that ∂Ω is smooth and, hence, that the boundary is a real-analytic (non-singular) curve. Observe that when ∂Ω is a real-analytic curve and h(z, ¯z) is real-analytic near ∂Ω, then there is a unique holomorphic function R(z) near ∂Ω such that h(z, ¯z) = R(z) on ∂Ω. Hence, another way of formulating the main hypothesis in this paper, under the a priori assumption that ∂Ω is a real-analytic curve and the boundary data is real-analytic, is that the unique holomorphic extension R(z) of h(z, ¯z) is a rational function. Since any simply connected bounded domain Ω is conformally equivalent to the unit disk D, we are naturally led to results involving the Riemann maps of Ω onto the unit disk D. Furthermore, in [5] it is shown that the Bergman kernel K(z, w) associated with Ω is rational (as a function of z and ¯w) if and only if any Riemann map is rational. Thus, we also obtain connections between the Dirichlet problem and the Bergman kernel, all of which are contained in our main theorem, Theorem 1. 2. Main results Our main results are summarized in the following theorem. Theorem 1. Let Ω be a simply connected bounded domain in C with smooth real-analytic boundary, and take distinct points a, a 1 , a 2 ∈ Ω. Let K(z, w) denote the Bergman kernel associated with Ω. Then the following are equivalent. (i) The solution to the Dirichlet problem (1) is rational for h(z, ¯z) = 1 z − a .
5 (2005), No. 2 The Dirichlet Problem with Rational Holomorphic Data 447 (ii) The solution to the Dirichlet problem (1) is rational for h(z, ¯z) = R(z), where R(z) is any rational function without poles on ∂Ω. (iii) The Riemann map f : Ω → D is rational. (iv) K(z, a 1 ) and K(z, a 2 ) are rational functions of z. (v) K(z, w) is a rational function of z and ¯w. We mention here a result by S. Bell [5] stating that the Bergman kernel on a multiply connected domain is rational if and only if the domain is simply connected and a Riemann map f : Ω → D is rational. This implies the equivalence (iii) ⇐⇒ (v) above. However, the proof we give is independent of Bell’s result. The proof proceeds as follows: We first prove a preliminary lemma, then prove the chain of implications (v) ⇒ (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) ⇒ (v). For clarity, we have written out some of these implications as theorems. We conclude this section with a few remarks concerning notation. Throughout the proof of Theorem 1 we shall use the notation g ∗ (z) := g(z), and shall use O(Ω) to denote the space of analytic functions in Ω. (Note that g is holomorphic in some domain Ω if and only if g ∗ is holomorphic in the domain Ω ∗ := {z : ¯z ∈ Ω}.) We shall use the notation P 1 for the Riemann sphere (a.k.a. the extended complex plane). We also use the notation C(z) for the field of rational functions in z. 3. Proof of Theorem 1 We start by proving the following simple lemma. Lemma 1. Suppose f is an algebraic function whose derivative is rational. Then f itself is rational. Proof. Let a 1 , a 2 , . . . , a n be the distinct poles of f ′ . Expand f ′ in terms of its singular parts: n∑ f ′ (z) = p(z) + s k (z) (2) = p(z) + = p(z) + k=1 n∑ p k ∑ k=1 j=1 p n∑ ∑ k k=1 j=2 c k,j (z − a k ) j c k,j n∑ (z − a k ) + j k=1 c k,1 z − a k , where the p k and c k,j are fixed constants, and p(z) is a polynomial.
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