Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets.
Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ $$u|_{\Gamma_1} = g$$ $$u|_{\Gamma_2} = 0.$$
How does one define and prove existence of weak solution for such a problem? Are there any references?
What I find in the literature when the boundary is disjoint is that one part of the boundary is given Dirichlet condition and the other is given Neumann condition but I don't want this.