Decompositions of functions and Dirichlet problems in the unit disc

Abstract

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.

Riemann–Hilbert problems with shift on the Lyapunov curve for null-solutions of iterated Beltrami equations

“…Subsequently, since these BVPs have a wide range of applications, such as singular integral equations, operator theory, partial differential equations (PDEs), shell theory, fluid dynamics, elasticity theory, computational mechanics, and so on, they were extensively studied by many scholars [1,[5][6][7][8][9]. In recent years, these problems (in particular, Riemann BVPs) have been generalized to ones for some different classes of functions like Cauchy-type integrals with densities from variable exponent Lebesgue space [10], polymonogenic functions in Clifford analysis [11], polyanalytic functions [12][13][14], polyharmonic functions [15][16][17], and even the families defined by other PDEs (especially polynomial Cauchy-Riemann equations) on some different curves with some suitable boundary value conditions [18][19][20][21][22].…”

LpPolyharmonic Dirichlet problems in regular domains I: the unit disc

“…Higher order Schwarz kernels are the key in our programme to solve the PHD problems (1.1). By the decomposition theorem of polyharmonic functions [8] and all above lemmas, we have the following…”

“…In recent years, the study of explicit solutions of BVPs has undergone a new phase of development ( [1-8, 10, 11, 14-16, 20] and references therein). These include Dirichlet, Neumann, Schwarz and Robin problems for harmonic, biharmonic, polyharmonic and polyanalytic equations in regular domains (in the unit disc: [1,2,4,7,8,11]; and in the upper-half plane: [3,5,6,10]) and in irregular domains (Lipschitz domains: [5,16,20]).…”

“…Proof. The property 3 indicates G n (z, t) is polyharmonic in H. By the decomposition theorem of polyharmonic functions in [8], (2.23) holds with (2.24)-(2.26) where G n,j (z, t) defined on H × R is analytic in H regarding to z and has at least the jth order zero at…”

Lp polyharmonic Robin problems on Lipschitz domains

“…There they found that the solutions can be represented by some integrals in terms of some kernel functions and tried to utilize complex integrals method to explicitly formulate the kernels. However their techniques to obtain kernel functions are complicated and indeed valid only for some small m. In [11], Du, Guo and Wang introduced some new ideas and got explicit expressions of the kernel functions in a unified way. From then on, these kernel functions are called higher order Poisson kernels because they are higher order analog of the classical Poisson kernel.…”