= Weak mixed Poisson problem
Let $U\subset\mathbb R^d$ be a bounded connected <Lipschitz domain>, and let a measurable boundary part $\Gamma_D$ have positive surface measure. Impose a homogeneous <Dirichlet boundary condition> there and a homogeneous <Neumann boundary condition> on the remainder. For $f\in L^2(U)$, seek $u$ in the closed <Sobolev space> subspace
$$
V=\{v\in H^1(U):\operatorname{Tr}v=0\text{ on }\Gamma_D\}
$$
satisfying
$$
\int_U\nabla u\cdot\nabla v=\int_Ufv\qquad(v\in V).
$$
The <Trace operator> makes the boundary restriction meaningful. The <Poincare inequality with a partial Dirichlet boundary> bounds the full $H^1$ <norm> by the gradient <norm> on $V$. Thus the left side is a bounded <coercive bilinear form>, and the <Lax-Milgram theorem> gives a unique <weak solution>. The same proof allows any bounded <linear functional> on $V$ as the right side.
The Neumann condition is natural: <integration by parts> leaves the boundary term on the non-Dirichlet portion, and requiring the identity for all $v\in V$ makes that term vanish. If $\Gamma_D$ is empty, constants remain in the <kernel of a linear map>; for homogeneous Neumann data, compatibility requires $\int_Uf=0$, and uniqueness requires a normalization such as zero mean. Positive Dirichlet surface measure is what removes this ambiguity.
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