= Weak mixed Poisson problem
Let a positive-measure part $\Gamma_D$ of the boundary of a bounded connected domain carry a homogeneous <Dirichlet boundary condition>, and let the complementary part carry a homogeneous <Neumann boundary condition>. On
$$
V=\{v\in H^1(U):\operatorname{Tr}v|_{\Gamma_D}=0\},
$$
the weak Poisson problem is
$$
\int_U\nabla u\mathbin\cdot\nabla v=\int_Ufv
\qquad(v\in V).
$$
The <Poincare inequality with a partial Dirichlet boundary> makes the left side coercive, so the <Lax-Milgram theorem> gives a unique weak solution.
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