Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H1(Ω) using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H1(Ω) does not apply, in fact the restriction of H1(Ω) functions is not necessarily in L2(∂Ω). So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H2(Ω) and we obtain an a priori estimate for the second derivatives of the solution.  2005 Elsevier Inc. All rights reserved.