Weak Dirichlet-to-Neumann map for bounded conductivities

Assume the Axiom of Choice and Countable Choice. Let $d\ge 2$, let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected smooth domain, and let $\gamma \in L^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ be an almost-everywhere class with $0<m\le \gamma \le M<\mathrm{\infty }$. Set $\mathcal{D}(\mathrm{\Omega }):=H^1(\mathrm{\Omega };\mathbb{C})\mathrm{/}{H}_0^1(\mathrm{\Omega };\mathbb{C})$ with quotient norm $\mathrm{\parallel }[U]{\mathrm{\parallel }}_{\mathcal{D}}:={\inf }_{w\in H_0^1(\mathrm{\Omega })}\mathrm{\parallel }U+w{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}$, and let $\mathcal{D}(\mathrm{\Omega }{)}^{\ast }$ be its continuous conjugate-linear dual. For $f=[U]\in \mathcal{D}(\mathrm{\Omega })$, let $u_{\gamma ,f}$ be the unique element of the coset $f$ satisfying ${\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }{u}_{\gamma ,f}\cdot \overline{\mathrm{\nabla }z}dx=0(z\in H_0^1(\mathrm{\Omega };\mathbb{C}))\mathrm{.}$ Define the weak Dirichlet-to-Neumann map ${\mathrm{\Lambda }}_{\gamma }:\mathcal{D}(\mathrm{\Omega })\to \mathcal{D}(\mathrm{\Omega }{)}^{\ast }$ by $\left({\mathrm{\Lambda }}_{\gamma }f\right)(g):={\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }{u}_{\gamma ,f}\cdot \overline{\mathrm{\nabla }{V}_g}dx,$ where $V_g$ is any representative of $g\in \mathcal{D}(\mathrm{\Omega })$. The trace quotient identification makes $\mathcal{D}(\mathrm{\Omega })$ the $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ boundary data space with its trace quotient norm. Constants lie in the kernel of ${\mathrm{\Lambda }}_{\gamma }$, and its range functionals annihilate constants; hence ${\mathrm{\Lambda }}_{\gamma }$ induces ${\dot{\mathrm{\Lambda }}}_{\gamma }:\mathcal{D}(\mathrm{\Omega })\mathrm{/}\mathbb{C}\mathbf{1}\to (\mathcal{D}(\mathrm{\Omega })\mathrm{/}\mathbb{C}\mathbf{1}{)}^{\ast }$, the quotient Dirichlet-to-Neumann map used for Calderón stability. Well-posedness, independence from the representative $V_g$, boundedness and the induced quotient map are established by the well-posedness result cited below, after translating and dilating the bounded domain into the unit cube if necessary. Such an affine map identifies the Sobolev spaces and zero-boundary cosets by bounded isomorphisms and multiplies both energy integrals by the same positive factor.

The real bounded conductivity class and uniform ellipticity bounds.

The trace quotient realization of the boundary $H^{1\mathrm{/}2}$ space.

The $H^1$ and $H_0^1$ notation.

The zero-boundary Sobolev closure.