- This paper studies the regularity of solutions to boundary value problems for the Laplace operator on Lipschitz domains in Rd and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in Lp() are the Besov spaces B (L()), := (/d + 1/p)-1. Thus, the regularity of the solution in this scale of Besov spaces is investigated with the aim of determining the largest for which the solution is in B(Lr()). The regularity theorems given in this paper build upon the recent results of Jerison and Kenig . The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions.