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 subspacesatisfyingThe 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 ${\int }_{U}f=0$, and uniqueness requires a normalization such as zero mean. Positive Dirichlet surface measure is what removes this ambiguity.