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

TL;DR

This paper introduces a new method using harmonic polynomials to uniformly approximate solutions to the Dirichlet problem for Laplace's equation on a disk, providing explicit convergence rates and improved estimates.

Contribution

It presents a novel approach demonstrating uniform convergence of harmonic polynomials for the Dirichlet problem, with explicit convergence rates and derivative estimates.

Findings

Harmonic polynomials converge uniformly to the solution

Smoother boundary data accelerates convergence

Improved constants and radius of convergence for Taylor series

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.