Green's Identities

Let $\Omega$ be a bounded Lipschitz domain with outward unit normal $\nu$.

Divergence theorem. For a vector field $\mathbf{F}$, ${\int }_{\Omega }\nabla \cdot \mathbf{F}\mathrm{d}x={\int }_{\partial \Omega }\mathbf{F}\cdot \nu \mathrm{d}s.$

Green’s first identity (integration by parts). Apply the divergence theorem to $v\kappa \nabla u$: ${\int }_{\Omega }v\nabla \cdot (\kappa \nabla u)\mathrm{d}x=-{\int }_{\Omega }\kappa \nabla u\cdot \nabla v\mathrm{d}x+{\int }_{\partial \Omega }v\kappa {\partial }_{\nu }u\mathrm{d}s.$

Green’s second identity (symmetric form): ${\int }_{\Omega }(v\nabla \cdot (\kappa \nabla u)-u\nabla \cdot (\kappa \nabla v))\mathrm{d}x={\int }_{\partial \Omega }\kappa (v{\partial }_{\nu }u-u{\partial }_{\nu }v)\mathrm{d}s.$

Uses in EIT.

• Deriving the Weak Formulation of the Conductivity Equation.
• Showing that the Dirichlet-to-Neumann Map is symmetric (second identity with two solutions; see Properties of the Boundary Operators).
• Deriving the linearisation identity and the Adjoint Equation.

General adjoint identity. For a linear differential operator $P$ with formal adjoint $P^{\ast }$: ${\int }_{\Omega }\lambda (Pv)={\int }_{\Omega }(P^{\ast }\lambda )v+B(\lambda ,v){|}_{\partial \Omega }$. The boundary form $B$ determines the boundary conditions of adjoint problems (see Rules of the Calculus of Variations).

References

1. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019