Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp |
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Authors: | Gabriel Acosta María G Armentano Ricardo G Durn Ariel L Lombardi |
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Affiliation: | aInstituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, B1613GSX Provincia de Buenos Aires, Argentina;bDepartamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina |
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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 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. |
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Keywords: | Cuspidal domains Traces Neumann problem Regularity |
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