Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint

Theorem
Let $\mathbf p$ be continuously differentiable.

The boundary conditions


 * $\map{\mathbf y} a\Big\vert_{x\mathop=a}=\map{\boldsymbol\psi} {\mathbf y}\Big\vert_{x\mathop=a}$

are self-adjoint :


 * $\forall i,k\in\N:1\le i,k\le N:\left.{\dfrac {\partial p_i \left[{x, \mathbf y, \boldsymbol \psi \left({\mathbf y}\right)}\right]} {\partial y_k} }\right\vert_{x \mathop = a} = \left.{\dfrac {\partial p_k \left[{x, \mathbf y, \boldsymbol \psi \left({\mathbf y}\right)}\right]} {\partial y_i} }\right\vert_{x \mathop = a}$

Necessary Condition
By assumption the boundary conditions are self-adjoint.

Then exists $\map g {x,\mathbf y}$ such that:


 * $\map{p_i} {x,\mathbf y,\map{\boldsymbol\psi} {\mathbf y} }=\frac{\partial \map g {x\mathbf y} } {\partial y_i}$

Since $\mathbf p\in C^1$, $g\in C^2$.

Differentiate both sides $y_k$:


 * $\frac{\partial\map{p_i} {x,\mathbf y,\map{\boldsymbol\psi} {\mathbf y} } } {\partial y_k} = \dfrac {\partial^2 g \left({x, \mathbf y}\right)} {\partial y_k \partial y_i}$

By the Schwarz-Clairaut Theorem, partial derivatives commute, hence indices can be mutually replaced:


 * $\dfrac{\partial\map {p_i} {x,\mathbf y,\map{\boldsymbol\psi} {\mathbf y} } } {\partial y_k} =\frac {\partial \map{p_k} {x,\mathbf y,\map{\boldsymbol\psi} {\mathbf y} } } {\partial y_i}$

Fixing $x=a$ provides the result.

Sufficient condition
By assumption:


 * $\displaystyle{\frac{\partial p_i} {\partial y_j}\Big\vert_{x\mathop=a}={\frac{\partial p_j} {\partial y_i}\vert_{x\mathop=a}$

Then


 * $\displaystyle\exists\map g {x,\mathbf y}\in C^2:\frac{\partial p_i}{\partial y_j}\Big\vert_{x\mathop=a}={\frac{\partial p_j} {\partial y_i} }\Big\vert_{x\mathop=a}=\frac{\partial^2 g} {\partial y_i\partial y_j}\Big\vert_{x\mathop=a}$

In other words:
 * $\displaystyle p_i\Big\vert_{x\mathop=a}=\frac{\partial g} {\partial y_i} \Big\vert_{x\mathop=a}$

Hence, the boundary conditions are self-adjoint.