18 problems

Well-definedness conjecture. The experimental conductivity $\gamma_{\mathrm{exp}}$ is well defined for every conductivity $\gamma\in\mathcal{G}^R$.

Let $(\mathcal M,\textsl{g})$ be a Lorentzian manifold satisfying the hypotheses (H1) and (H3), and consider the unique continuation theorem in the exterior of the double null cone…

Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds with mutual boundary, and let $\Lambda_{g_1}$ and $\Lambda_{g_2}$ denote their Dirichlet-to-Neumann maps for harmonic functi…

Let $n \geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Suppose that $\gamma_1,\gamma_2 \in W^{1,n}(\Omega)$ are uniformly elliptic, and let…

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, let $\gamma$ be the conductivity, and assume $d\geq 3$. The Dirichlet-to-Neumann data determine $\gamma$ uniquely. Brown's conj…

Higher-dimensional nonuniqueness conjecture. The map $\gamma\mapsto\Lambda_\gamma$ is not one-to-one in higher dimensions, in particular for $N=3$.

Modified anisotropic Calderón conjecture at nonzero frequency. If

Anisotropic Calderón conjecture at zero frequency. If

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, and let $W^{1,d}(\Omega)$ denote the Sobolev space of functions with one weak derivative in $L^d(\Omega)$. Optimal-regularity c…

Let $d/2\leq p<\infty$, let $\Omega$ be a bounded domain with Lipschitz boundary, and let $H_c^{s,p}(\Omega)$ denote the compactly supported Sobolev space of potentials. For a pote…

Anisotropic partial Calderón conjecture. It should follow that $g=\tilde{g}$ up to the diffeomorphism gauge invariance if $\dim M\geq 3$, and up to the diffeomorphism and conformal…

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary. Let $f \in C^{\infty}(M)$ vanish to infinite order at $\partial M$ and satisfy … for all $u_j \in C^{\infty}(M)$…

Uniqueness conjecture for the Schrödinger potential. The knowledge of $\Lambda_{g,q}$ uniquely determines $q$. This is the potential-recovery version of the Calderón problem in a k…

Let $(M,g)$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $A$ and $B$ be unitary connections on $E$. Denote by $\Lambda_A$ and $\Lambda_B$ the Di…

Uhlmann's conjecture. Lipschitz conductivities should be uniquely determined by their Dirichlet-to-Neumann maps $\Lambda_\gamma$.

Two-point determination conjecture. If both

Uhlmann's conjecture. The optimal regularity assumption for uniqueness in Calderón's inverse conductivity problem is that the conductivities are Lipschitz.

Let $(M,g_1)$ and $(M,g_2)$ be two compact oriented Riemannian manifolds with smooth boundary and dimension $n\geq 3$. For a metric $g$, let … be the Cauchy data set of harmonic fu…