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

• Published in: Journal of mathematical analysis and applications Vol. 310; no. 2; pp. 397 - 411
• Main Authors: Acosta, Gabriel, Armentano, María G., Durán, Ricardo G., Lombardi, Ariel L.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15-10-2005; Elsevier
• Subjects: Cuspidal domains; Exact sciences and technology; Mathematical analysis; Mathematics; Neumann problem; Partial differential equations; Regularity; Sciences and techniques of general use; Traces; Neumann problem; Traces; Regularity; Cuspidal domains; Trace; Solution uniqueness; Mathematical analysis; Solution regularity; Poisson equation; Cuspidal domain; Cusp
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.01.065

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.