Calderón admissible conductivity classes

Fix an integer $d\ge 2$. Let $\mathrm{\Omega }\subset (0,1{)}^d$ be a nonempty bounded connected open domain with $C^{\mathrm{\infty }}$ boundary, and let ${\mathrm{\Omega }}^{\mathrm{\prime }}\subset \mathrm{\Omega }$ be an open domain whose closure is compact and contained in $\mathrm{\Omega }$. Fix real parameters $m>0$, $M\ge \max \{1,m\}$, $R>0$, and $0<\alpha <1$. Define ${\mathrm{\Gamma }}_{m,M}(\mathrm{\Omega }):=\{\gamma \in L^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R}):m\le \gamma (x)\le M\text{ for almost every }x\in \mathrm{\Omega }\}$ and ${\mathrm{\Gamma }}_{m,M}^{\mathrm{\prime }}(\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}):=\{\gamma \in {\mathrm{\Gamma }}_{m,M}(\mathrm{\Omega }):\gamma =1\text{ almost everywhere on }\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\mathrm{\prime }}\}\mathrm{.}$ For $\gamma \in BV(\mathrm{\Omega })$ set $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{BV(\mathrm{\Omega })}:=\max \{\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{L^1(\mathrm{\Omega })},TV(\gamma ;\mathrm{\Omega })\}$. For a continuously differentiable function on $\bar{\mathrm{\Omega }}$ whose first derivatives are $\alpha$-Hölder continuous, set $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{C^{1,\alpha }(\bar{\mathrm{\Omega }})}:=\max \{{\max }_{\mathrm{\mid }\beta \mathrm{\mid }\le 1}{\sup }_{x\in \bar{\mathrm{\Omega }}}\mathrm{\mid }{\mathrm{\partial }}^{\beta }\gamma (x)\mathrm{\mid },{\max }_{\mathrm{\mid }\beta \mathrm{\mid }=1}{\sup }_{\begin{array}{c}x,y\in \bar{\mathrm{\Omega }}\\ x\mathrm{\ne }y\end{array}}\frac{\mathrm{\mid }{\mathrm{\partial }}^{\beta }\gamma (x)-{\mathrm{\partial }}^{\beta }\gamma (y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }}\}\mathrm{.}$ Define $X_d(R):=\{\begin{array}{ll}\{\gamma \in {\mathrm{\Gamma }}_{m,M}^{\mathrm{\prime }}(\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }})\cap BV(\mathrm{\Omega }):\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{BV(\mathrm{\Omega })}\le R\},& d=2,\\ \{\gamma \in {\mathrm{\Gamma }}_{m,M}^{\mathrm{\prime }}(\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }})\cap C^{1,\alpha }(\bar{\mathrm{\Omega }}):\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{C^{1,\alpha }(\bar{\mathrm{\Omega }})}\le R\},& d\ge 3.\end{array}$ For $\gamma \in {\mathrm{\Gamma }}_{m,M}^{\mathrm{\prime }}(\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }})$, define its measurable background extension ${\gamma }_e:{\mathbb{R}}^d\to \mathbb{R}$ by ${\gamma }_e=\gamma$ almost everywhere on $\mathrm{\Omega }$ and ${\gamma }_e=1$ on ${\mathbb{R}}^d\setminus \mathrm{\Omega }$. Then ${\gamma }_e-1$ vanishes almost everywhere outside ${\mathrm{\Omega }}^{\mathrm{\prime }}$. No pointwise continuity of ${\gamma }_e$ is asserted. The classes $X_d(R)$ are not assumed nonempty.

Spatial BV classes and total variation.

Lebesgue spaces are almost-everywhere classes.