Lax-Milgram Theorem

Theorem (Lax–Milgram). Let $V$ be a Hilbert space, $a:V\times V\to \mathbb{R}$ a bilinear form that is

• bounded: $|a(u,v)|\le C\Vert u{\Vert }_V\Vert v{\Vert }_V$, and
• coercive: $a(u,u)\ge \alpha \Vert u{\Vert }_V^2$ with $\alpha >0$,

and $\ell \in V^{\ast }$. Then there is exactly one $u\in V$ with $a(u,v)=\ell (v)$ for all $v\in V$, and $\Vert u{\Vert }_V\le \Vert \ell {\Vert }_{V^{\ast }}/\alpha$.

Application to EIT. Take $a(u,v)={\int }_{\Omega }\gamma \nabla u\cdot \nabla v$ with $0<{\gamma }_{\min }\le \gamma \le {\gamma }_{\max }$.

• Boundedness: $|a(u,v)|\le {\gamma }_{\max }\Vert \nabla u{\Vert }_{L^2}\Vert \nabla v{\Vert }_{L^2}$.
• Coercivity: $a(u,u)\ge {\gamma }_{\min }\Vert \nabla u{\Vert }_{L^2}^2$. This controls the full $H^1$ norm on
  • $V=H_0^1(\Omega )$ by the Poincaré inequality (Dirichlet Problem), and
  • $V=H^1(\Omega )/\mathbb{R}$ by the Poincaré–Wirtinger inequality (Neumann Problem); see Sobolev and Trace Spaces.
• Right-hand side: $\ell (v)={\int }_{\partial \Omega }gv$ is bounded on $H^1$ by the trace theorem if $g\in H^{-1/2}(\partial \Omega )$. It is well defined on the quotient space iff $\int g=0$.

So the forward problems are well-posed, with the stability estimate $\Vert \nabla u{\Vert }_{L^2}\le C\Vert g{\Vert }_{H^{-1/2}}/{\gamma }_{\min }$. The same argument applies to the adjoint equation (see Adjoint Equation). The lower bound ${\gamma }_{\min }>0$ is essential, which is one reason for Box Constraints on Conductivity.

References

1. P. D. Lax, A. N. Milgram (1954). Parabolic equations. In: Contributions to the Theory of Partial Differential Equations, Ann. of Math. Stud. 33, 167–190. doi:10.1515/9781400882182-010
2. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019