Corner compatibility condition for mixed boundary data ($g^{\prime }(b)=q(d)$)

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 $g^{\prime }(b)=q(d)$: differentiating the top boundary trace in its tangential $x$ direction gives $u_x(b,d)=g^{\prime }(b)$, 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 $g^{\prime }(b)=0$ for a solution continuously differentiable at the top-right corner. Boundary values may remain continuous even when this derivative requirement fails.