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Vector transform operators for piece-wise harmonic functions

O. Yaremko, Y. Parfenova

TL;DR

The paper develops vector transform operators as a generalization of scalar transform methods to solve Laplace-type boundary value problems in piecewise homogeneous, spherically symmetric domains. It constructs matrix-valued kernels $H_{k,s,l}^*$ and associated operators $P_0$, $P_{jq}$ to map solutions across internal interfaces, yielding explicit representations that satisfy outer and interfacial conditions. Using the influence-function approach, it derives radial ODEs for each angular mode on the sphere and builds integral representations of the solutions in terms of boundary data and matrix kernels, with concrete realizations for a third vector boundary-value problem in the unit circle and a Dirichlet problem. Additionally, transform methods in the half-plane (via $J$ and $J^{-1}$) are discussed, leveraging eigenfunctions of coupled Sturm–Liouville problems. The framework aims to extend to configurations with multiple internal interfaces, offering a structured approach for multi-layer harmonic problems.

Abstract

The vector transform operators are investigated; these operators are used at the solution of boundary value problems in piecewise homogeneous spherically symmetric areas. In particular, examples of transformation operators for vector boundary value problems are given for third vector boundary value problem in the unit circle and for the Dirichlet problem in the unit circle.

Vector transform operators for piece-wise harmonic functions

TL;DR

The paper develops vector transform operators as a generalization of scalar transform methods to solve Laplace-type boundary value problems in piecewise homogeneous, spherically symmetric domains. It constructs matrix-valued kernels and associated operators , to map solutions across internal interfaces, yielding explicit representations that satisfy outer and interfacial conditions. Using the influence-function approach, it derives radial ODEs for each angular mode on the sphere and builds integral representations of the solutions in terms of boundary data and matrix kernels, with concrete realizations for a third vector boundary-value problem in the unit circle and a Dirichlet problem. Additionally, transform methods in the half-plane (via and ) are discussed, leveraging eigenfunctions of coupled Sturm–Liouville problems. The framework aims to extend to configurations with multiple internal interfaces, offering a structured approach for multi-layer harmonic problems.

Abstract

The vector transform operators are investigated; these operators are used at the solution of boundary value problems in piecewise homogeneous spherically symmetric areas. In particular, examples of transformation operators for vector boundary value problems are given for third vector boundary value problem in the unit circle and for the Dirichlet problem in the unit circle.

Paper Structure

This paper contains 6 sections, 1 theorem, 61 equations.

Table of Contents

  1. Introduction.
  2. The common boundary value problem for the Laplase equation in unit sphere with non-uniform internal conditions of interface.
  3. Method of influence function.
  4. Transform operators
  5. Transform operators in half-plane
  6. Conclusion.

Key Result

Theorem 1

If the conditions of existence i)-ii) vector-valued function $H_{k,s,l}^{*} \left(r,\rho \right),$ satisfied, then transformation operator $P_{0}$$\left(P_{jq} \right)$ compare $\hat{u}_{0}$ harmonic vector function $\left(\hat{u}_{jq} \right),$ in the $B_{0}$ homogeneous ball and piesewise harmonic

Theorems & Definitions (1)

  • Theorem 1