EIT forward continuity and compactness of conductivity classes

If $m>1$, the fixed positive-measure boundary collar $\mathrm{\Omega }\setminus \overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ makes $\gamma =1$ incompatible with $\gamma \ge m$ almost everywhere. Thus $X_d(R)=\mathrm{\varnothing }$; it and its kernel image are compact, and the pairwise estimates and continuity assertions are vacuous (one may take $q=3$ and $C=1$). Henceforth assume $0<m\le 1\le M$, as required by [F7]. In dimension two, set $h_{\gamma }=\mathrm{zeroextend}(\gamma -1)$. It is supported in $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$, has $\mathrm{\parallel }{h}_{\gamma }{\mathrm{\parallel }}_1\le (M+1)\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$, and satisfies $\mathrm{\parallel }{\tau }_y{h}_{\gamma }-h_{\gamma }{\mathrm{\parallel }}_1=\mathrm{\parallel }{\tau }_y{\gamma }_e-{\gamma }_e{\mathrm{\parallel }}_1\le C\mathrm{\mid }y\mathrm{\mid }$ by [F1]. Apply the Fréchet–Kolmogorov criterion [F2] to the bounded-support family $\{h_{\gamma }\}$ in $L^1({\mathbb{R}}^2)$; restriction to $\mathrm{\Omega }$ followed by addition of one gives relative compactness of $X_2(R)$ in $L^1(\mathrm{\Omega })$. If ${\gamma }_j\to \gamma$ in $L^1$, pass to an almost-everywhere convergent subsequence; the ellipticity bounds and background-collar equality pass to the limit almost everywhere. For each admissible smooth test field $\phi$, its divergence is bounded, so the pairing with ${\gamma }_j$ converges to the pairing with $\gamma$. By [F5], the absolute value of each approximating pairing is at most $TV({\gamma }_j;\mathrm{\Omega })$; taking the supremum over $\phi$ yields $V(\gamma )\le {\mathrm{lim\hspace{0.17em}inf}}_{j}TV({\gamma }_j;\mathrm{\Omega })$. Assumption [A3] lets [F6] infer $\gamma \in BV(\mathrm{\Omega })$ and $TV(\gamma ;\mathrm{\Omega })=V(\gamma )\le R$. Also $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_1={\lim }_j\mathrm{\parallel }{\gamma }_j{\mathrm{\parallel }}_1\le R$, so the defining maximum BV norm is at most $R$. Thus the class is closed and compact in $L^1$.

For $d\ge 3$, the uniform $C^{1,\alpha }$ bound gives uniform bounds and equicontinuity for both the functions and their first derivatives. Apply [F3] with [A4] to the finite vector $({\gamma }_j,D{\gamma }_j)$ to extract a subsequence converging in $C^1(\bar{\mathrm{\Omega }})$. The limit derivatives retain the same $\alpha$-Hölder bound by passing to the limit in $\mathrm{\mid }D{\gamma }_j(x)-D{\gamma }_j(y)\mathrm{\mid }\le R\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$; the ellipticity and collar conditions are closed under uniform convergence. Hence $X_d(R)$ is compact in $C^1$.

Apply [F7], using the observation convention [F4], with the actual fixed smoothing exponent $r_d$. The Hilbert–Schmidt difference there has, by the kernel isometry in the smoothing lemma, exactly the norm $\mathrm{\parallel }{\kappa }_{\gamma }-{\kappa }_{{\gamma }_0}{\mathrm{\parallel }}_2$. In dimension two this gives the stated $L^q$ Lipschitz estimate with $q>2$. In dimensions at least three its supremum-norm estimate gives continuity in $C^1(\bar{\mathrm{\Omega }})$, since convergence in that topology implies uniform convergence of the coefficients. Thus the asserted kernel continuity is established in both branches.

To use step 1.1's $L^1$ compactness in dimension two, note that all conductivities have the same uniform $L^{\mathrm{\infty }}$ bound. Hence $\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_q^q\le (2M{)}^{q-1}\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_1$. The estimate in step 1.3 therefore makes the kernel map continuous also for the $L^1$ topology on this class. Its image is compact as the continuous image of the compact class from step 1.1. For $d\ge 3$, use the $C^1$ compactness in step 1.2 and the continuity in step 1.3. This proves the final assertion without a finite-dimensional measurement restriction.