In this thesis, we apply the method of layer potentials to prove that there is a unique solution to the Dirichlet problem for Laplace's equation in C^{1}-domains. We assume that C^{1}-domains are subsets of R^{d}, d≥ 2, they are bounded and they have connected boundaries. In addition, we assume that the boundary data of the Dirichlet problem belong to the Lebesgue space L^{p}(∂ D,σ) with p∈(1,∞).
We will follow the work of E. B. Fabes, M. Jodeit Jr. and N. M. Rivi{\`e}re. In their work, they solved various boundary value problems in domains that were merely C^{1} by applying the method of layer potentials. The method of layer potentials is a procedure for solving the boundary value problems in the form of layer potentials. We will use it to solve the Dirichlet problem in the form of the double layer potential. The double layer potential satisfies Laplace's equation and the boundary values of the double layer potential are given by an operator \tfrac{1}{2}I+K. It turns out that in C^{1}-domains the operator K is compact on L^{p}(∂ D). Consequently, we can use the Fredholm theory to deduce that the operator \tfrac{1}{2}I+K is invertible and thus, we obtain a double layer potential solution to the Dirichlet problem. Finally, we will establish the uniqueness of the solution by using the properties of Green's function.
In the end of this thesis, we will also discuss how the method of layer potentials can be applied to the Dirichlet problem for Laplace's equation in Lipschitz domains with L^{2}-boundary data by following the doctoral dissertation of G. H. Verchota.
