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

Gabriel Acosta,María G Armentano,Ricardo G Durán,Ariel L Lombardi

doi:10.1016/j.jmaa.2005.01.065

Abstract

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 H 1 ( 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 H 1 ( Ω ) does not apply, in fact the restriction of H 1 ( Ω ) functions is not necessarily in L 2 ( ∂ 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 H 2 ( Ω ) and we obtain an a priori estimate for the second derivatives of the solution.

Highlights

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
To apply the Lax-Milgram theorem in this case one needs to use some trace theorem for Sobolev spaces
In [6] the authors characterize the traces of the Sobolev spaces W 1,p(Ω), 1 ≤ p < ∞ for domains of the class considered here by using some weighted norm on the boundary

Summary

Introduction

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. We consider the following model problem: let Ω be the plane domain defined by In order to obtain existence results for more general data we present a different kind of trace results by introducing a weighted Sobolev space in Ω such that the restriction to the boundary of functions in that space are in Lp(Γ).

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Results

Conclusion