ABSTRACT.

Let \( D \) be a connected bounded domain in \( \mathbb{R}^{n} \), \( n \ge 2 \), \( S \) be its boundary, which is closed and smooth. Consider the Dirichlet problem

\[
\Delta u = 0 \ \text{in } D, \qquad u|_{S} = f,
\]

where \( f \in L^{1}(S) \) or \( f \in H^{-\ell} \), where \( H^{-\ell} \) is the dual space to the Sobolev space \( H^{\ell} := H^{\ell}(S) \), \( \ell \ge 0 \) is arbitrary.

The aim of this paper is to prove that the above problem has a solution for an arbitrary \( f \in L^{1}(S) \) and this solution is unique and to prove similar results for rough (distributional) boundary values. These results are new. The method of its proof, based on the potential theory, is also new. Definition of the \( L^{1}(S) \)-boundary value and a distributional boundary value of a harmonic function in \( D \) is given. For \( f \in L^{1}(S) \), the difficulty comes from the fact that the product of an \( L^{1}(S) \) function times the kernel of the potential on \( S \) is not absolutely integrable.

We prove that an arbitrary \( f \in H^{-\ell} \), \( \ell > 0 \), can be the boundary value of a harmonic function in \( D \).