Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. Consider the following problem \begin{cases} -\Delta u = \lambda u & \mbox{in }\Omega\\ u=0 & \mbox{in }\Gamma^c\\ \partial_\nu u=0 & \mbox{in }\Gamma \end{cases} where obviously $\Gamma^c$ is the complementar of $\Gamma$ in $\partial\Omega$, s.t. $\partial\Omega = \bar\Gamma \cup \bar\Gamma^c$ and $\Gamma \cap \Gamma^c =\emptyset$. What is the better functional space to define a variational formulation for this Mixed Dirichlet-Neumann problem? More precisely the space $$ H^1_0(\Omega\cup \Gamma) = \mbox{closure of } \mathcal{C}^1_0(\Omega\cup\Gamma) \mbox{ with respect to the norm of } H^1(\Omega) $$ can be studied and considered as an usual $H^1_0(D)$ with $D\subset\mathbb{R}^N$, in order to use the general results associated to the eigenvalue problem with Dirichlet boundary condition, or the fact $\Omega\cup\Gamma$ is not open can obstruct this idea?

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The space you mention is the right one. See the notes at the end of chapter 8 of the book of Gilbarg and Trudinger. If I remember correctly such mixed boundary problems are treated in the book by Duvaut and Lions on Inequalities in Mechanics and Physics. Mixed Dirichlet-Neumann problems are often referred to as Zaremba problems.