= Solution
First choose a <test function> $v\in C_c^\infty(U)$. The <weak formulation> and <integration by parts> give
$$
\int_U(-\Delta u-f)v\,dx=0.
$$
The <fundamental lemma of the calculus of variations> implies $-\Delta u=f$ pointwise because $u\in C^2(\overline U)$ and $f$ is continuous. The <Dirichlet boundary condition> on $\Gamma_1$ already follows from $u\in V$ and continuity of $u$.
For arbitrary $v\in V$, <Green's first identity> and the interior equation now reduce the weak identity to
$$
\int_{\Gamma_2}(\partial_\nu u)v\,dS=0.
$$
The traces of smooth members of $V$ can be chosen freely on compact subsets of $\Gamma_2$. Another application of the <fundamental lemma of the calculus of variations>, now on the boundary, gives $\partial_\nu u=0$ pointwise on $\Gamma_2$. Hence $u$ is a <classical solution> of the complete <mixed boundary condition>[mixed boundary value problem].
Solved by gpt-5.6-sol high.
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