= Coercive weak formulation of a coupled Dirichlet-Neumann system
{c}
For $H=H_0^1(U)\times H^1(U)$, the coupled form
$$
B((u,w),(v,z))=\int_U\bigl(Du\mathbin\cdot Dv+uv+wv+Dw\mathbin\cdot Dz+wz-3uz\bigr)
$$
satisfies
$$
B((u,w),(u,w))=\|Du\|_2^2+\|Dw\|_2^2+\|u-w\|_2^2.
$$
The <Poincaré inequality> for $u$ then controls both $\|u\|_2$ and $\|w\|_2$, so the form is <coercive> on $H$. The <Lax-Milgram theorem> therefore gives a unique <weak solution>. The first component has a <Dirichlet boundary condition>, while the second component's <Neumann boundary condition> is the natural boundary condition in the <weak formulation>.
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