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Uniform approximation by harmonic polynomials for solving the Dirichlet problem of Laplace's equation on a disk

Haesung Lee

TL;DR

This work tackles the Dirichlet problem for Laplace's equation on the disk by constructing a solution as a uniform limit of harmonic polynomials. It develops a polar-coordinate reformulation, leverages Fourier series and Jackson's theorem to obtain explicit convergence rates, and then transfers the construction back to Cartesian coordinates to produce explicit harmonic polynomials with provable uniform convergence on the disk. The approach yields sharp, order-preserving bounds for high-order derivatives in terms of $L^1$ boundary data and demonstrates analyticity of the solution inside the disk, along with an improved radius of convergence for Taylor expansions at every interior point. Overall, the paper provides a self-contained, Poisson-formula-free framework that delivers quantitative convergence, smoothness, and analytic structure for the Dirichlet problem in a disk with strong practical and theoretical implications.

Abstract

In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution using harmonic polynomials that converge uniformly to a solution. Specifically, we rigorously derive the convergence rate of the harmonic polynomials and show that smoother boundary data and proximity of the target point to the disk's origin accelerate the convergence. Additionally, we obtain uniform estimates for the derivatives of solutions of arbitrary orders, controlled by $L^1$-boundary data. Notably, the constants in our estimates are significantly improved compared to existing results. Furthermore, we provide an enhanced radius of convergence for Taylor's series of the solution at each point in the open disk.

Uniform approximation by harmonic polynomials for solving the Dirichlet problem of Laplace's equation on a disk

TL;DR

This work tackles the Dirichlet problem for Laplace's equation on the disk by constructing a solution as a uniform limit of harmonic polynomials. It develops a polar-coordinate reformulation, leverages Fourier series and Jackson's theorem to obtain explicit convergence rates, and then transfers the construction back to Cartesian coordinates to produce explicit harmonic polynomials with provable uniform convergence on the disk. The approach yields sharp, order-preserving bounds for high-order derivatives in terms of boundary data and demonstrates analyticity of the solution inside the disk, along with an improved radius of convergence for Taylor expansions at every interior point. Overall, the paper provides a self-contained, Poisson-formula-free framework that delivers quantitative convergence, smoothness, and analytic structure for the Dirichlet problem in a disk with strong practical and theoretical implications.

Abstract

In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution using harmonic polynomials that converge uniformly to a solution. Specifically, we rigorously derive the convergence rate of the harmonic polynomials and show that smoother boundary data and proximity of the target point to the disk's origin accelerate the convergence. Additionally, we obtain uniform estimates for the derivatives of solutions of arbitrary orders, controlled by -boundary data. Notably, the constants in our estimates are significantly improved compared to existing results. Furthermore, we provide an enhanced radius of convergence for Taylor's series of the solution at each point in the open disk.
Paper Structure (11 sections, 17 theorems, 116 equations)

This paper contains 11 sections, 17 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Solving the problem in the polar coordinates
  4. Coordinate transformations
  5. Solutions in the polar coordinate
  6. Fourier series and Jackson's theorem
  7. Constructing a solution in the Cartesian coordinates
  8. Constructing harmonic polynomials
  9. Uniform convergence of harmonic polynomials on $\overline{B}_R$
  10. Constructing a solution in $C^{\infty} (B_R)\cap C^\alpha(\overline{B}_R)$
  11. Acceleration of uniform convergence

Key Result

Theorem 1.1

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 4.1
  • Theorem 4.2
  • Proposition 4.3: Summation by parts
  • Theorem 4.4
  • Theorem 4.5
  • ...and 9 more