Jorge L. deLyra1
Department of Mathematical Physics,
Physics Institute,
University of São Paulo
March 19, 2018
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.