Properties of the Boundary Operators

Let $u_f$ solve the Dirichlet Problem with data $f$. Green’s first identity (see Green’s Identities) applied to $\nabla \cdot (\gamma \nabla u_f)=0$ gives

$$⟨{\Lambda }_{\gamma }f,h⟩={\int }_{\partial \Omega }h\gamma {\partial }_{\nu }{u}_f\mathrm{d}s={\int }_{\Omega }\gamma \nabla u_f\cdot \nabla u_h\mathrm{d}x.$$

Several properties of the Dirichlet-to-Neumann Map ${\Lambda }_{\gamma }$ follow from this symmetric form:

1. Self-adjoint: $⟨{\Lambda }_{\gamma }f,h⟩=⟨f,{\Lambda }_{\gamma }h⟩$.
2. Positive semidefinite: $⟨{\Lambda }_{\gamma }f,f⟩={\int }_{\Omega }\gamma |\nabla u_f{|}^2\ge 0$, which is the dissipated power.
3. Kernel = constants: equality holds iff $\nabla u_f=0$, that is, $f$ is constant.
4. Zero-mean range: taking $h=1$ gives ${\int }_{\partial \Omega }{\Lambda }_{\gamma }f=0$.
5. Homogeneity: ${\Lambda }_{c\gamma }=c{\Lambda }_{\gamma }$ for constants $c>0$.
6. Monotonicity: ${\gamma }_1\le {\gamma }_2$ a.e. implies ${\Lambda }_{{\gamma }_1}\le {\Lambda }_{{\gamma }_2}$ as quadratic forms. This follows from the Dirichlet principle and is the basis of monotonicity-based inclusion detection.
7. Nonlinearity: in general ${\Lambda }_{{\gamma }_1+{\gamma }_2}\ne {\Lambda }_{{\gamma }_1}+{\Lambda }_{{\gamma }_2}$ (see Forward Map).

The Neumann-to-Dirichlet Map inherits the corresponding properties on zero-mean functions, with the monotonicity reversed: ${\gamma }_1\le {\gamma }_2\Rightarrow {\mathcal{R}}_{{\gamma }_1}\ge {\mathcal{R}}_{{\gamma }_2}$.

Because ${\mathcal{R}}_{\gamma }$ is self-adjoint and positive, its discretisation has an orthogonal eigendecomposition $\mathcal{R}=Q\Sigma Q^{\top }$ with $\Sigma \ge 0$. This is used for Truncated SVD Regularization and for choosing Current Patterns.

References

1. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
2. D. Gisser, D. Isaacson, J. C. Newell (1990). Electric Current Computed Tomography and Eigenvalues. SIAM J. Appl. Math. 50(6), 1623–1634. doi:10.1137/0150096
3. B. Harrach, M. Ullrich (2013). Monotonicity-based shape reconstruction in electrical impedance tomography. SIAM J. Math. Anal. 45(6), 3382–3403. doi:10.1137/120886984