Logarithmic EIT stability on explicit conductivity classes — Examples and Counterexamples

By [F2] with radii $1\mathrm{/}2<1$, a bump $\psi$ as in the Example exists; it is nonzero since it equals one on $B(0,1\mathrm{/}2)$. Its first and second derivatives are bounded because it is smooth with compact support. If $\mathrm{\mid }x-y\mathrm{\mid }\le 1$, the mean-value estimate gives $\mathrm{\mid }\mathrm{\nabla }\psi (x)-\mathrm{\nabla }\psi (y)\mathrm{\mid }\le \mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y\mathrm{\mid }\le \mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$; if $\mathrm{\mid }x-y\mathrm{\mid }\ge 1$, the quotient is at most $2\mathrm{\parallel }\mathrm{\nabla }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus $0<N_{\alpha }<\mathrm{\infty }$ and the normalization yields $0\mathrm{\ne }\varphi$ with $\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{C^{1,\alpha }}\le 1$.

The support of $x\mapsto \varphi ((x-x_0)\mathrm{/}r)$ lies in $B(x_0,r)\subset D^{\mathrm{\prime }}$, so $\gamma =1$ on $D\setminus D^{\mathrm{\prime }}$. Since $\mathrm{\mid }\varphi \mathrm{\mid }\le 1$ and $\epsilon \le 1\mathrm{/}4$, its values lie in $[3\mathrm{/}4,5\mathrm{/}4]\subset [1\mathrm{/}2,3\mathrm{/}2]$.

The sup norm of the perturbation is at most $\epsilon \le 1\mathrm{/}4$. Its first derivative suprema are at most $\epsilon \mathrm{/}r\le r^{\alpha }\mathrm{/}8\le 1\mathrm{/}8$, and the $\alpha$-Hölder seminorm of each first derivative is at most $\epsilon r^{-1-\alpha }[\mathrm{\nabla }\varphi {]}_{C^{\alpha }}\le 1\mathrm{/}8$ by scaling the Hölder quotient. The derivatives of the constant background vanish, so $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{C^{1,\alpha }(\bar{D})}\le 5\mathrm{/}4<2$. All conditions in [F1] hold, hence $\gamma \in X_d(2)$.

The disk $B(x_0,1\mathrm{/}16)$ is contained in $D^{\mathrm{\prime }}$ because $1\mathrm{/}16<1\mathrm{/}8$, and $\gamma =1$ on $D\setminus D^{\mathrm{\prime }}$. Almost everywhere its only values are $1$ and $3\mathrm{/}2$, so $1\mathrm{/}2\le \gamma \le 3\mathrm{/}2$. Thus the support and ellipticity conditions in [F1] hold.

Since $\mathrm{\mid }D\mathrm{\mid }=\pi \mathrm{/}16$ and $\mathrm{\mid }B(x_0,1\mathrm{/}16)\mathrm{\mid }=\pi \mathrm{/}256$, the integral satisfies $\mathrm{\parallel }\gamma {\mathrm{\parallel }}_{L^1(D)}=\pi \mathrm{/}16+\pi \mathrm{/}512=33\pi \mathrm{/}512<1$. By [F2] and homogeneity of distributional variation, $TV(\gamma ;D)\le \frac{1}{2}(2\pi \mathrm{/}16)=\pi \mathrm{/}16<1$; restriction to $D$ cannot increase the full-space variation. Therefore the BV norm is at most $1$, so [F1] gives $\gamma \in X_2(1)$.

At every point of the circle $\mathrm{\partial }B(x_0,1\mathrm{/}16)$ lying inside $D$, every neighborhood meets the interior disk and its exterior in sets of positive measure. The function equals $3\mathrm{/}2$ almost everywhere on the former and $1$ almost everywhere on the latter. A continuous representative would be constant $3\mathrm{/}2$ on the interior and constant $1$ on the exterior by continuity and almost-everywhere equality, contradicting continuity across the circle. Hence the conductivity class is discontinuous.

For every $j\ge 1$, $r_j\le a\mathrm{/}8<a\mathrm{/}2$, so the inclusion lies in $D^{\mathrm{\prime }}$. Its values are $1$ and $3\mathrm{/}2$, and its $L^1(D)$ norm is $\pi a^2+\frac{1}{2}\pi r_j^2\le \pi \mathrm{/}16+\pi \mathrm{/}2048<1$. By [F4], $TV({\gamma }_j;D)=\frac{1}{2}(2\pi r_j)=\pi r_j\le \pi a\mathrm{/}8<1$. Also ${\gamma }_0$ equals the background and has $L^1(D)=\pi \mathrm{/}16<1$ and zero variation. Thus [F1] gives ${\gamma }_0,{\gamma }_j\in X_2(1)$.

The squared $L^2(D)$ difference is ${\int }_{B(x_0,r_j)}(1\mathrm{/}2{)}^{2}dx=\pi r_j^2\mathrm{/}4$, hence its norm is $\frac{1}{2}\sqrt{\pi }{r}_j$. Apply [F2] with $\beta =1\mathrm{/}2$, so $\eta =(1\mathrm{/}2)\mathrm{/}(5\mathrm{/}2)=1\mathrm{/}5$ and $2\eta \mathrm{/}(1-\eta )=1\mathrm{/}2$; this gives the stated DN bound. The well-posedness and definition facts [F3] ensure the operator comparison is in the stated quotient-space norm.

If the asserted finite constant $K$ existed, step 1.1 puts both coefficients in its class, while step 1.2 computes the data and DN differences; hence the estimate applied to $({\gamma }_j,{\gamma }_0)$ would give $\frac{1}{2}\sqrt{\pi }{r}_j\le \frac{K}{2}(r_j\mathrm{/}a{)}^2$. Dividing by $r_j>0$ yields $\sqrt{\pi }\le K{r}_j\mathrm{/}{a}^2$ for every $j$. Since $r_j\to 0$, the right-hand side tends to zero, contradicting $\sqrt{\pi }>0$. Therefore the refuted statement is false.