The exercise is the exercise 6.6 from Evans' PDE.
Suppose $U$ is connected, and $\partial U$ consists of two disjoint, closed sets $\Gamma_1$ and $\Gamma_2$. Define what it means for $u$ to be a weak solution of Poisson equation with mixed Dirichlet-Neumann boundary conditions:
$$
\begin{cases}
-\Delta u = f \ \ \ \text{in $U$} \\
u = 0 \ \ \ \text{on $\Gamma_1$} \\
\frac{\partial u}{\partial \nu} = 0 \ \ \text{on $\Gamma_2$}.
\end{cases}
$$
My attempts: Let $u \in C^{\infty}(U)$ be a solution to the above problem. Then for $v \in H^1(U)$, integration by parts yields
\begin{align}
(f,v) = -\int_U (\Delta u) v & = \int_U Du \cdot Dv -\int_{\partial U} \frac{\partial u}{\partial \nu} v \\
& = \int_U Du\cdot Dv – \int_{\Gamma_1} \frac{\partial u}{\partial \nu} v
\end{align}
I wish to conclude $\int_{\Gamma_1} \frac{\partial u}{\partial \nu} v = 0$, but I don't know how to do that. Could anyone give me some hint?
Best Answer
You can pick the space of test functions $v$ to be the space $$H^1_{\Gamma_1}(U) = \{v \in H^1(U)\colon v = 0 \text{ on } \Gamma_1 \}, $$ which is a Hilbert space with respect to the inner product in $H^1$. Actually, you can verify that in $H^1_{\Gamma_1}$ the norm $$ \|v\|_{H^1_{\Gamma_1}(U)} = \|Dv\|_{L^2(U)}$$ is equivalent to the $H^1$ norm (by Poincaré inequality). Then the second integral equals $0$ since $v \in H^1_{\Gamma_1}(U).$