L^pPolyharmonic Dirichlet problems in regular domains I: the unit disc

Abstract

In this article, we consider a class of Dirichlet problems with L p boundary data for polyharmonic functions in the upper half plane. By introducing a sequence of new kernel functions called higher order Schwarz kernels, integral representation solutions of the problems are given.1991 Mathematics Subject Classification. 30G30. Key words and phrases. Polyharmonic functions, Dirichlet problems, higher order Schwarz kernels, integral representation.The first named author is partially supported by the NNSF grant (No. 10871150) and greatly appreciates Professors Dr. Jinyuan Du, Heinrich Begehr and Tao Qian for many things.where all limits are non-tangential [19]. Definition 2.2. Let D be a simply connected (bounded or unbounded) domain in the plane with smooth boundary ∂D, and H(D) denote the set of all analytic functions in D. If f is a continuous function defined on D × ∂D satisfying f (•, t) ∈ H(D) for any fixed t ∈ ∂D, and f (z, •) ∈ L p (∂D), p ≥ 1, (or C 0 (∂D) if ∂D is a boundless curve) for any fixed z ∈ D, then f is H × L p (H × C 0 ) on D × ∂D and write f ∈ (H × L p )(D × ∂D) (or (H × C 0 )(D × ∂D)). Likewise, (H × C)(D × ∂D) may be similarly defined. Lemma 2.3. Let D be a simply connected unbounded domain in the plane with smooth boundary ∂D. If f is defined on D × ∂D that is analytic in D for any fixed t ∈ ∂D and (2.1) |f (z, t)| ≤ M 1 |t − z | uniformly on D c × {t ∈ ∂D : |t| > T } whenever z ∈ D c , where D c is any compact set in D, and M, T are positive constants depending only on D c , then for any fixed z 0 ∈ D, the primitive function (2.2) F (z, t) = z z0 f (ζ, t)dζ, z ∈ D, t ∈ ∂D, enjoys the same properties as f. Proof. It is trivial to show the analyticity of F (z, t) in z ∈ D for each fixed t ∈ ∂D ( [18]). Then the inequality part of the Lemma follows from (2.3) |F (z, t)| ≤ γ [z 0 ,z] |f (ζ, t)||dζ| since γ [z0,z] ∩ D c is compact for any simple curve γ [z0,z] in D that connects z 0 ∈ D with z ∈ D c .

Lp polyharmonic Robin problems on Lipschitz domains

“…From then on, these kernel functions are called higher order Poisson kernels because they are higher order analog of the classical Poisson kernel. Then Du and his collaborators also obtained higher order Poisson kernels of some regular domains such as the unit ball, the upper half plane and space, and applied higher order Poisson kernels to solve the corresponding polyharmonic Dirichlet problems (that is, the above PHD problem) in these regular domains ( [12][13][14][15]). Furthermore, in [10], excepting the above polyharmonic Dirichlet problems with L p boundary data, Du also studied the polyharmonic Neumann and regularity problems on bounded Lipschitz domains and Lipschitz graph domains as follows:…”