Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane

Abstract

In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that correspondence lead to very general proofs of existence of solutions of the Dirichlet problem on the plane. At first, this establishes the existence of solutions for almost arbitrary integrable real functions on the unit circle, including functions which are discontinuous and unbounded. The proof of existence is then generalized to a large class of non-integrable real functions on the unit circle. Further, the proof of existence is generalized to real functions on a large class of other boundaries on the plane, by means of conformal transformations.

26 pgs. Small formatting corrections and bibliography update

References in corpus (7)

• Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
• Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
• Fourier Theory on the Complex Plane V: Arbitrary-Parity Real Functions, Singular Generalized Functions and Locally Non-Integrable Functions
• Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
• Complex Analysis of Real Functions II: Singular Schwartz Distributions
• Complex Analysis of Real Functions III: Extended Fourier Theory
• Complex Analysis of Real Functions IV: Non-Integrable Real Functions

Cited by in corpus (1)

• Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem