Given the following elliptic differential equation: $$ - \nabla \cdot (\lambda \nabla u ) = f$$ in $\Omega$ $$ \lambda \nabla u \cdot n = b - au $$ on $\partial \Omega$
where $\Omega$ is a bounded field with smooth edge and $\lambda, f, a, b$ smooth functions with $ \lambda \geq \lambda_0 > 0$ and $a \geq a_0 > 0$.
Is there any equivalent minimization problem to this boundary value problem?
Taking the notion that DMan mentioned above, then we can rewrite this problem as, allowing $f = \nabla\cdot g$, $$ \|\lambda \nabla u+g\|_2^2. $$
Stationarity of this problem implies that (as a heuristic derivation without lagrange multipliers for the boundary conditions, which can be included later), $$ 0 = 2\nabla\cdot (\lambda \nabla u + g) = 2(\lambda \nabla^2 u + \nabla\cdot g) \implies -\lambda \nabla^2 u = \nabla\cdot g = f, $$ which is roughly what we wanted. This problem can then be written as $$ \begin{equation*} \begin{aligned} & \underset{u}{\text{minimize}} & & \|\lambda \nabla u+g\|_2^2 \\ & \text{subject to} & & \lambda \nabla u\cdot n = b-au, \text{ over } \partial\Omega. \end{aligned} \end{equation*} $$
The latter constraint is linear, and the functional above is strongly convex and smooth, so everything is nice and has the expected solutions. The equivalence between this problem and the one stated can be noted via stationarity conditions of the problem (and strong duality), while uniqueness of the solution follows from strong convexity everywhere in the domain.