 
Summary: NONHOMOGENEOUS NEUMANN PROBLEM FOR THE POISSON
EQUATION IN DOMAINS WITH AN EXTERNAL CUSP
GABRIEL ACOSTA, MAR´IA G. ARMENTANO, RICARDO G. DUR´AN, AND ARIEL L. LOMBARDI
Abstract. In this work we analyze the existence and regularity of the solution of a nonho
mogeneous Neumann problem for the Poisson equation in a plane domain with an external
cusp.
In order to prove that there exists a unique solution in H1
() using the LaxMilgram theorem
we need to apply a trace theorem. Since is not a Lipschitz domain, the standard trace theorem
for H1
() does not apply, in fact the restriction of H1
() functions is not necessarily in L2
().
So, we introduce a trace theorem by using weighted Sobolev norms in .
Under appropriate assumptions we prove that the solution of our problem is in H2
() and
we obtain an a priori estimate for the second derivatives of the solution.
1. introduction
This paper deals with an elliptic equation in a domain with an external cusp. Since this kind
of domains are not Lipschitz, the standard arguments to prove existence can not be applied when
