Dirichlet Problem with L1(S) Boundary Values

Abstract

Let D be a connected bounded domain in R2, S be its boundary, which is closed and C2-smooth. Consider the Dirichlet problem Δu=0inD,u|S=h, where h∈L1(S). The aim of this paper is to prove that the above problem has a solution for an arbitrary h∈L1(S), and this solution is unique. The result is new. The method of its proof is new. The definition of the L1(S)-boundary value of a harmonic in the D function is given. No embedding theorems are used. The history of the Dirichlet problem goes back to 1828. The result in this paper is, to the author’s knowledge, the first result in the 194 years of research (since 1828) that yields the existence and uniqueness of the solution to the Dirichlet problem with the boundary values in L1(S).