= Corner compatibility condition for mixed boundary data
{title2=$g'(b)=q(d)$}
Consider the top-right corner $(b,d)$ of a rectangle. Suppose the top <Dirichlet boundary condition> is $u(x,d)=g(x)$ and the right <Neumann boundary condition>, with outward normal in the positive $x$ direction, is $u_x(b,y)=q(y)$. A solution that is a <continuously differentiable function> up to the corner necessarily satisfies $g'(b)=q(d)$: differentiating the top <boundary trace> in its tangential $x$ direction gives $u_x(b,d)=g'(b)$, while the right-edge condition gives the same <partial derivative> as $q(d)$.
This is a necessary compatibility condition, not an existence theorem. If it fails, a solution can still satisfy the interior equation and the boundary conditions on open edges while failing to have a continuous first derivative at that corner. In particular, homogeneous right-edge <Neumann boundary conditions> require $g'(b)=0$ for a solution continuously differentiable at the top-right corner. Boundary values may remain continuous even when this derivative requirement fails.
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