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Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data

  • *Corresponding author: Jesse Railo

    *Corresponding author: Jesse Railo 

The first author is supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters

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  • We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain $ \Omega \subset {\mathbb R}^n $ and any disjoint open sets $ W_1, W_2 \Subset {\mathbb R}^n \setminus \overline{\Omega} $ there always exist two positive, bounded, smooth, conductivities $ \gamma_1, \gamma_2 $, $ \gamma_1 \neq \gamma_2 $, with equal partial exterior Dirichlet-to-Neumann maps $ \Lambda_{\gamma_1}f|_{W_2} = \Lambda_{\gamma_2}f|_{W_2} $ for all $ f \in C_c^{\infty}(W_1) $. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property $ \gamma_i^{1/2}-1 \in H^{2s, \frac{n}{2s}}( {\mathbb R}^n) $ for $ i = 1, 2 $. We also provide counterexamples on domains that are bounded in one direction when $ n \geq 4 $ or $ s \in (0, n/4] $ when $ n = 2, 3 $ using a modification of the argument on bounded domains.

    Mathematics Subject Classification: Primary: 35R30; Secondary: 26A33, 42B37, 46F12.


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  • Figure 1.  A graphical illustration of a possible geometric setting of Theorem 1.1. Here $ \Omega\subset {\mathbb R}^n $ is taken for simplicity to be a bounded domain and the measurements are performed in the disjoint nonempty open subsets $ W_1, W_2\subset \Omega_e $. Moreover, the supports of the background deviations $ m_1, m_2 $ do not necessarily coincide in the exterior $ \Omega_e $ and can even be disjoint as illustrated above. In this case uniqueness may be lost

    Figure 2.  A graphical illustration of the sets used in the proof of Theorem 1.2. Here $ \Omega\subset {\mathbb R}^n $ is an arbitrary open bounded set and the measurements are performed in the disjoint open sets $ W_1, W_2\subset\Omega_e $. In the proof, we construct in the first step a nonzero $ s $-harmonic background deviation $ \tilde{m}_2\in H^s( {\mathbb R}^n) $ in the set $ \Omega' $, which has a smooth boundary and lies in the deformed annulus $ \Omega_{3\epsilon}\setminus\overline{\Omega}_{2\epsilon} $, and then obtain by mollification a nonzero smooth $ s- $harmonic function $ m_2: = \tilde{m}_2\ast \rho_{\epsilon} $ in the set $ \Omega $. The set $ \omega\Subset\Omega_e\setminus\overline{W_1\cup W_2} $ is used to construct a cut-off function $ \eta\in C_c^{\infty}(\omega_{3\epsilon}) $ with $ \eta|_{\overline{\omega}} = 1 $, which $ \tilde{m}_2 $ has as an exterior value, and in the end this function leads to the property that $ m_2 $ is nonnegative and has compact support contained in $ \Omega_{5\epsilon}\cup \omega_{5\epsilon} $. Note that the topology of the sets $ \Omega $ and $ \omega $ could be also very complicated

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