Lagrangian Formulation

For the constraint “u solves the Neumann Problem with conductivity $\sigma$, current $g$ and interior source $h$”, the Lagrangian couples the objective with the constraint through a multiplier $\lambda$ (the adjoint state):

$$\mathcal{L}(\sigma ,u,\lambda )=J(u)+⟨\lambda ,e(\sigma ,u)⟩.$$

Strong form. $e(\sigma ,u)=\nabla \cdot (\sigma \nabla u)+h$ gives $\mathcal{L}=J(u)+{\int }_{\Omega }\lambda (\nabla \cdot (\sigma \nabla u)+h)\mathrm{d}x$.

Weak form (preferred). Integrating by parts once and inserting $\sigma {\partial }_{\nu }u=g$ gives

$$\mathcal{L}(\sigma ,u,\lambda )=J(u)-{\int }_{\Omega }\sigma \nabla u\cdot \nabla \lambda \mathrm{d}x+{\int }_{\Omega }h\lambda \mathrm{d}x+{\int }_{\partial \Omega }g\lambda \mathrm{d}s.$$

Advantages of the weak form:

1. The Neumann datum $g$ appears explicitly.
2. Every variation needs at most one integration by parts. Terms with ${\partial }_{\nu }\delta u$, which have no trace for $\delta u\in H^1$, never appear.
3. $\sigma$ appears only in one term, which is linear in $\sigma$, so the $\sigma$-variation is exact and produces no boundary terms.

Function spaces. Trial space $V$ for $u$ and space $V_0$ of admissible variations:

forward problem	$V$	$V_0$
Dirichlet, $u=f$	$\{v\in H^1:v{|}_{\partial \Omega }=f\}$	$H_0^1(\Omega )$
Neumann, $\sigma {\partial }_{\nu }u=g$	$H^1(\Omega )/\mathbb{R}$	$H^1(\Omega )/\mathbb{R}$

The multiplier lives in $V_0$. For the Dirichlet problem drop the boundary integral, since $\lambda \in H_0^1$.

Setting the variations to zero gives the State Equation, the Adjoint Equation and the functional derivative (see KKT Conditions and Adjoint State Method).

References

1. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1
2. A. M. Bradley (2024). PDE-constrained optimization and the adjoint method. Lecture notes, Stanford. cs.stanford.edu/~ambrad/adjoint_tutorial.pdf