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Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp. (English) Zbl 1073.35073

Summary: We analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain \(\Omega\) with an external cusp. In order to prove that there exists a unique solution in \(H^{1}(\Omega)\) using the Lax-Milgram theorem we need to apply a trace theorem. Since \(\Omega\) is not a Lipschitz domain, the standard trace theorem for \(H^{1}(\Omega)\) does not apply, in fact the restriction of \(H^{1}(\Omega)\) functions is not necessarily in \(L^{2}(\partial \Omega)\). So, we introduce a trace theorem by using weighted Sobolev norms in \(\Omega\). Under appropriate assumptions we prove that the solution of our problem is in \(H^{2}(\Omega)\) and we obtain an a priori estimate for the second derivatives of the solution.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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References:

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