Consider the mixed Dirichlet-Neumann BVP \begin{equation*} \left\{ \begin{alignedat}{2} {} (-\Delta) u & {}= f && \quad\mbox{ in } \, \Omega; \\ u & {}= \phi && \quad\mbox{ in }\,D;\\ \partial_\nu u & {}= \psi && \quad\mbox{ in }\,N, \end{alignedat} \right. \end{equation*}

where D and N are Dirichlet and Neumann subsets of $\Omega.$ For this problem, I have some questions:

1) What are the different approaches to solve mixed Dirichlet Neumann problem?

2) How do we approach it by Variational formulation specially with Non homogeneous Dirichlet-Neumann BVP?

3) What are the appropriate Sobolev space to deal with?

  • $\begingroup$ Choose $v \in H^{1}$ such that $v|_{\Gamma_D}=0$ and in order to apply the weak formulation multiply by $v$ both sides, integrate on $\Omega$ and use Green's identities, remember also that $\partial \Omega=\Gamma_D \cup \Gamma_N$ when you apply divergence theorem. Note that for this problem you have to require $\Gamma_D \ne \emptyset$ $\endgroup$ – VoB Jul 5 at 12:12
  • $\begingroup$ exactly, this will work if we have homogeneous Dirichlet boundary conditions. Do you have any idea how to work for non-homogeneous boundary conditions? I tried it by defining the Sobolev space using trace of $u$ to be $\phi$, but there is a problem when we put $u$ as a test function to get neumann condition! $\endgroup$ – R.Arora Jul 6 at 19:20

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