# Oleg Yu. Imanuvilov's research while affiliated with Colorado State University and other places

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## Publications (87)

We consider initial boundary value problems with the homogeneous Neumann boundary condition. Given an initial value, we establish the uniqueness in determining a spatially varying coefficient of zeroth-order term by a single measurement of Dirichlet data on an arbitrarily chosen subboundary. The uniqueness holds in a subdomain where the initial val...

We establish the Lipshitz stability estimate in inverse problem of determination of a source term or zero order term in the Schr\"odinger equation with time-dependent coefficients under some non-trapping assumption. Based on this result we established the Lipshitz stability of the determination of a real-valued coefficient corresponding to zero-th...

For linearized Navier–Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at t0>0 over the spatial domain. The key machinery are Carleman estimates for the Navier–Stokes equations and the op...

For linearized Navier-Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at $t_0>0$ over the spatial domain. The key are Carleman estimates for the Navier-Stokes equations and the operator...

For linearized Navier-Stokes equations, we first derive a Carleman estimate with a regular weight function. Then we apply it to establish conditional stability for the lateral Cauchy problem and finally we prove conditional stability estimates for inverse source problem of determining a spatially varying divergence-free factor of a source term.

For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t , we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. A...

In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.

For an initial-boundary value problem for a parabolic equation in the spatial variable $x=(x_1,.., x_n)$ and time $t$, we consider an inverse problem of determining a coefficient which is independent of one spatial component $x_n$ by extra lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse...

In this article, we provide a modified argument for proving stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method does not require any cut-off procedures and therefore simplifies the existing proofs. We establish the conditional stability for inverse source problems for a...

In this article, we discuss the methodology based on Carleman estimates concerning the unique continuation and inverse problems of determining spatially varying coefficients. First as retrospective views we refer to main works by that methodology starting from the pioneering work by Bukhgeim and Klibanov published in 1981. Then as one possible obje...

In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and can simplify the existing proofs. We establish the conditional stability for inverse source problems for a hyp...

We prove the exact controllability result to trajectories of a simplified model of motion of a rigid body in fluid flow. Unlike a previously know results such a trajectory does not need to be a stationary solution.

We consider the Kelvin-Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply the Carleman estimate to prove the Lipschitz stability in determining a spatial varying function in an external source term of Kelvin-Voigt model by a single measurement. Finally as a related system, we con...

We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of viscoelasticity system. Since a solution $u$ under consideration is not assumed to have compact support, in the decoupling of the Lam\'e operator by introducing div $u$ and rot $...

In this note, we prove that for the Navier–Stokes equations, a pair of Dirichlet and Neumann data and pressure uniquely correspond to a pair of Dirichlet data and surface stress on the boundary. Hence the two inverse boundary value problems in Imanuvilov and Yamamoto (2015 Inverse Probl.
31 035004) and Lai et al (Arch. Rational Mech. Anal.) are th...

For the two-dimensional Schrödinger equation in a bounded domain, we prove uniqueness of the determination of potentials in Wp1(Ω), p >. 2 in the case where we apply all possible Neumann data supported on an arbitrarily non-empty open set Γ~ of the boundary and observe the corresponding Dirichlet data on Γ~. An immediate consequence is that one can...

We consider inverse boundary value problems for the Schrodinger equations in
two dimensions. Within less regular classes of potentials, we establish a
conditional stability estimate of logarithmic order. Moreover we prove the
uniqueness within $L^p$-class of potentials with $p > 2$.

We consider inverse boundary value problems for the Navier–Stokes equations and the isotropic Lamé system in two dimensions. The question of global uniqueness for these inverse problems, without any smallness assumptions on unknown coefficients, has been a longstanding open problem for the Navier–Stokes equations and the isotropic Lamé system in tw...

We prove the global uniqueness in determination of the conductivity, the permeability and the permittivity of the two-dimensional Maxwell equations by the partial Dirichlet-to-Neumann map limited to an arbitrary subboundary.

We prove some uniqueness results in determination of the conductivity, the permeability and the permittivity of Maxwell's equations in a cylindrical domain Omega x (0, L) from partial boundary map. More specifically, for an arbitrarily given subboundary Gamma(0) subset of partial derivative Omega, we prove that the coefficients of Maxwell's equatio...

We consider a parabolic system with [Inline formula] components in a bounded spatial domain [Inline formula] over a time interval [Inline formula] whose principal part is coupled and discuss the backward problem in time of determining initial data [Inline formula], [Inline formula] from [Inline formula], [Inline formula]. We prove two conditional s...

We consider an inverse boundary value problem for diffusion equations with
multiple fractional time derivatives. We prove the uniqueness in determining a
number of fractional time-derivative terms, the orders of the derivatives and
spatially varying coefficients.

For Maxwell's equations in a wave guide, we prove the global uniqueness in
determination of the conductivity, the permeability and the permittivity by
partial Dirichlet-to-Neumann map limited to an arbitrary subboundary.

We consider an inverse problem of determining a spatially varying factor in a source term in the non-stationary linearized Navier–Stokes equations by observation data in an arbitrarily fixed sub-domain over some time interval. We prove the Lipschitz stability provided that the t-dependent factor satisfies a non-degeneracy condition. Our proof is ba...

We consider inverse boundary value problems for the Navier-Stokes equations
and the isotropic Lam\'e system in two dimensions. The uniqueness without any
smallness assumptions on unknown coefficients, which is called global
uniqueness, was longstanding open problems for the Navier-Stokes equations and
the isotropic Lam\'e system in two dimensions....

We consider inverse boundary value problems for elliptic equations of second
order of determining coefficients by Dirichlet-to-Neumann map on subboundaries,
that is, the mapping from Dirichlet data supported on $\partial\Omega\setminus
\Gamma_-$ to Neumann data on $\partial\Omega\setminus \Gamma_+$. First we prove
uniqueness results in three dimens...

For the Lamé system, we prove in the three-dimensional case that
both Lamé coefficients are uniquely recovered from partial Cauchy
data on an arbitrary open subset of the boundary provided that the
coefficient μ is a constant.

Let $\Omega\subset \Bbb R^2$ be a bounded domain with $\partial\Omega\in
C^\infty$ and $L$ be a positive number. For a three dimensional cylindrical
domain $Q=\Omega\times (0,L)$, we obtain some uniqueness result of determining
a complex-valued potential for the Schr\"odinger equation from partial Cauchy
data when Dirichlet data vanish on a subboun...

We consider an inverse problem of determining coefficient matrices in an
$N$-system of second-order elliptic equations in a bounded two dimensional
domain by a set of Cauchy data on arbitrary subboundary. The main result of the
article is as follows: If two systems of elliptic operators generate the same
set of partial Cauchy data on an arbitrary s...

We relax the regularity condition on potentials of Schr\"odinger equations in
the uniqueness results in \cite{EB} and \cite{IY2} for the inverse boundary
value problem of determining a potential by Dirichlet-to-Neumann map.

For the isotropic Lamé system we prove that if the Lamé coefficient μ is a positive constant, both Lamé coefficients may be recovered from the partial Cauchy data.

We discuss the inverse boundary value problem of determining the conduc-tivity in two dimensions from the pair of all input Dirichlet data supported on an open subset Γ + and all the corresponding Neumann data measured on an open subset Γ − . We prove the global uniqueness under some additional geometric condition, in the case where Γ + ∩ Γ − = ∅,...

Exact controllability of a multilayer plate system with free bound-ary conditions are obtained by the method of Carleman estimates. The multi-layer plate system is a natural multilayer generalization of a three-layer "sand-wich plate" system due to Rao and Nakra. In the multilayer version, m shear deformable layers alternate with m + 1 layers model...

We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollar...

We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural obstructions such as conformal invariance, gauge transform...

We prove for a two-dimensional bounded domain that the Cauchy data for
the Schroedinger equation measured on an arbitrary open subset of the
boundary uniquely determines the potential. This implies, for the
conductivity equation, that if we measure the current fluxes at the
boundary on an arbitrary open subset of the boundary produced by voltage
po...

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary
conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic
equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes s...

In this article, by a Carleman estimate for solutions of the two-dimensional non-stationary Lamé system with the stress boundary condition, we obtain the conditional stability estimate in determining the density and two Lamé coefficients which vary spatially by a single observation in a neighbourhood of a suitable subboundary. Moreover, we prove an...

We consider a hyperbolic differential operator with variable principal term. We first give a sufficient condition for the pseudoconvexity which yields a Carleman estimate and a necessary condition. Our sufficient condition implies that level sets generated by the weight function in the Carleman estimate are convex with respect to the set of rays gi...

We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage po...

We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a simply connected domain. The proof is reduced to show a similar result for the Schr\"odinger equation. Using...

We study the question of global controllability for the two-dimensional Burgers equation when the control acts on a part Γ 1 of the boundary Γ. We prove global controllability when Γ 1 is the whole boundary or in a specific geometrical situation when Γ 0 =Γ∖Γ 1 is contained in a parallel to the first bisector line. We also show with a counterexampl...

In this paper, for functions without compact supports, we established Carleman estimates for the two-dimensional non-stationary Lamé system with the stress boundary condition.

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove t...

For distributed controls we get a local exact controllability for the 2-D Navier-Stokes equations in the case where the fluid
is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

In this paper we deal with some controllability problems for systems of the Navier–Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control N-dimensional systems (N+1 scalar unknowns in the case of the Navier–Stokes equations) with N-1 scalar control functions. In a first step, we present some global...

In this paper we will discuss recent developments in controllability of evolution equations of fluid mechanics. The control is assumed to be distributed either on a part of the boundary or locally distributed in some subdomain. We will present some ideas of proof of main theorems. Special attention will be paid to the technique based on Carleman es...

In a bounded domain Ω ⊂ ℝn, we consider a hyperbolic operator P with the principal term ∂t2 - p(x, t)Δ. Under the assumption that the outer normal derivative of p is non-positive, we will estimate u in U × (-t0, t0 by the Cauchy data on an open subset of ∂Ω × (-T, T), where t 0 < T is some constant and U is a neighbourhood of ∂Ω. The condition on t...

In this paper, we establish Carleman estimates for the two
dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary
conditions. Using this estimate, we prove the uniqueness and the
stability in determining spatially varying density and two Lamé
coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is...

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllabil...

For the isotropic stationary Lamé system with variable coefficients equipped with the Dirichlet or surface stress boundary condition, we obtain a Carleman estimate such that (i) the right hand side is estimated in a weighted L 2-space and (ii) the estimate includes nonhomogeneous surface displacement or surface stress. Using this estimate we establ...

We prove Carleman inequalities for a second order parabolic equation when the coefficients are not bounded and norms of right hand sides are taken in the Sobolev space $L^2(0,T;W_2^{-\ell}(\Omega))$, $\ell\in [0,1]$. Our Carleman inequality yields the unique continuation for $L^2$-solutions. We further apply these inequalities to the global exact z...

For the solution u(p) = u(p)(t, x) to ∂t2 u(t, x) − div(p(x)∇u(t, x)) = 0 in (0, T) × Ω with given u|(0,T)×∂Ω, u(0, ) and ∂t u(0, ), we consider an inverse problem concerning the determination of the coefficient p(x), x Ω from data u|(0,T)×ω. Here Ω ⊂ n is a bounded domain, and ω is some subdomain of Ω and T > 0. For suitable ω ⊂ Ω and T > 0, we pr...

In this paper we construct a non-topological multivortex solution of a generalized version of the relativistic self-dual
Chern-Simons-Higgs system in that makes the energy functional finite. Our method of proof is an extension of the previous argument used by the authors
to prove the existence of general type of non-topological multivortex solution...

We consider a general second order elliptic equation with right-hand side where and Dirichlet boundary condition g∈H1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L² norms of f and fj and the H1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large...

We prove a Carleman inequality for the second order hyperbolic equation in the cylinder Q with the norm of righthand side taken in the space (W12(Q))*.

We study the existence and various behaviors of topological multivortex solutions of the relativistic self-dual Maxwell-Chern-Simons-Higgs system. We first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional. Then, by an iteration method, we prove the existence of topolo...

We consider weak solutions of a general second-order elliptic equation with right-hand side $$f+{\sum }_{j=0}^{N}\partial {f}_{j}/\partial {x}_{j}\in {H}^{-1}\left(\Omega \right)$$ with f , f j ∈ L 2(Ω) and boundary trace g ∈ H 1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L 2-norms of f an...

For the solution u (p) = u (p) (x, t) to partial derivative (2)(t)u, (x, t) -Deltau (x, t) - p(x)u(x, t) = 0 in Omega x (0, T) and partial derivativeu/partial derivativev\ partial derivative Omega x (0, T) = 0 with given u(., 0) and partial derivative (t)u (., 0), we consider an inverse problem of determining p(x), x epsilon Omega, from data u(\ome...

We consider an inverse problem of determining p(x), x ∈ Ω in (∂2 u/∂t 2)(x, t) − Δu(x, t) − p(x)u(x, t) = 0 in Ω × (0, T) and (∂u/∂ν)|∂Ω×(0, T) = 0 with given u(·, 0) and (∂u/∂t)(·, 0). Here Ω ⊂ R n , n = 1, 2, 3, is a bounded domain. We prove the Lipschitz stability in determining p from u|∂Ω×(0,T), under the assumption that T is greater than the...

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain $\Omega$ with control distributed in a subdomain $\omega\subset\Omega\subset \mathbb{R}^n, n\in\{2,3\}$. The result that we obtained in this paper is as follows. Suppose that $\hat v(t,x)$ is a given soluti...

This article deals with Neumann boundary optimal control problems associated with the Boussinesq equations including solid media. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations i...

This paper deals with optimal control problems associated with the 2-D Boussinesq equations. The controls considered may be of either the distributed or the Neumann type. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justifie...

An optimal control problem for a model for stationary, low Mach
number, highly nonisothermal, viscous flows is considered.
The control problem involves the minimization of a measure of
the distance between the velocity field and a given target
velocity field. The existence of solutions of a boundary value
problem for the model equations is establis...

We consider a Stokes system with external force term:
$$\frac{{\partial y}}{{\partial t}} = \Delta y - \nabla p + r\left( t \right)f\left( x \right),\nabla \cdot y = 0in\left( {0,T} \right) \times \Omega $$ and $${y_{\left| {\left( {0,T} \right) \times \partial \Omega } \right.}} = 0,$$
where Ω ⊂ ℝn
, n = 2,3 is a bounded domain. We discuss an inve...

. We construct a general type of multivortex solutions of the selfduality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic Chern-Simons model with the non-topological boundary condition near infinity. For such construction we use a modified version of the Newton iteration method developed by Kantorovich. Introduction The Lagr...

In this paper we first prove a new criterion of global solvability for smooth axisymmetric solutions of the 3-D Euler equations for a cylindrical type of domain, which can be whole R3. Then, using this result, we obtain generic solvability of the equations in the cylindrical domain which does not include the axis of symmetry. By a similar method we...

In this paper we are devoted to proving the existence and nonexistence of self-dual equations arising in Chern-Simons-Higgs theory with a constant electric charge density. There are three kinds of boundary conditions that admit solitonic structures. It is shown that there exist solutions in two cases of them. In the other case, we prove that there...

We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity $omega_0$, we assumed that $omega_0/r$ belongs to $L(log L (Bbb R^3))^{alpha}$ with $alpha >1/2$, where $r$ is the distance to an axis of symmetry. To prove the existence...

We consider a system
with a suitable boundary condition, where is a bounded domain, is a uniformly elliptic operator of the second order whose coefficients are suitably regular for , is fixed, and a function satisfies
Our inverse problems are determinations of g using overdetermining data or , where and . Our main result is the Lipschitz stabilit...

We study the local exact boundary controllability problem for the Boussinesq equations that describe an incompressible fluid flow coupled to thermal dynamics. The result that we get in this paper is as follows: suppose that y ^(t,x) is a given solution of the Boussinesq equation where t∈(0,T), x∈Ω, Ω is a bounded domain with C ∞ -boundary ∂Ω. Let y...

We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a
bounded domain with control distributed in an arbitrary fixed subdomain. The result that we obtain in this paper is as follows.
Suppose that we have a given stationary point of the Navier-Stokes equations and our initial...

For distributed controls we get a local exact controllability for the 2-D Boussinesq equations in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

In this paper we prove global-in-time existence and uniqueness of a positive solution for the system of nonlinear partial differential equations arising from an electrochemistry model. The powers of nonlinearity are allowed to be arbitrary positive integers, and our domain is any bounded subdomain of ℝ2 with a smooth boundary.

We study the local exact boundary controllability problem for the incompressible Navier-Stokes equations. The result is as follows: suppose that y ^(t,x) is a given solution of the Navier-Stokes equations where t∈(O,T), x∈Ω, Ω is a bounded domain with C ∞ -boundary ∂Ω. Let y 0 (x) be a given initial condition and ∥y ^(0,·)-y 0 ∥<ε where ε=ε(y ^) is...

This paper studies the problem of exact boundary controllability of second-order semilinear parabolic equations when the control is under Neumann's boundary conditions. For a nonlinear term with a sublinear growth we prove the global null-controllability and for the superlinear growth case we prove the local exact controllability of the equations.

Approximate controllability of the Stokes system is established by a constructive method when control is a right-hand-side concentrated in subdomain i.e. in the case of local distributed control. Approximate uncontrollability of the Burgers equation is shown in the cases of boundary and local distributed controls. A local theorem of exact controlla...

For two-dimensional Navier-Stokes equations defined in a bounded domain Omega and for an arbitrary initial vector field, we construct the boundary Dirichlet condition that is tangent to the boundary of partial derivative Omega of Omega and satisfies the property: the solution v(t, x) of the mentioned boundary-value problem equals zero at a certain...

In this paper, we establish Carleman estimates for the three-dimensional isotropic non-stationary Lame system with the homogeneous Dirichlet boundary con- ditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lame coefficients by a single measurement of solu- tion over (0 ,T ) × ω,...

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lame system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lame coefficients by a single measurement of solution over (0 ,T )×ω, where T> 0 is a...

In this lecture we present our recent results on the constructions of non-topological multivortex solutions of various self-dual Chern-Simons-Higgs systems in ℝ 2 which makes the energy functional finite. Our method of proof is basically the Newton-Kantorovich iteration. Thus, not only we prove existence of solutions, but also we present a method o...

## Citations

... In this article, it is clear that we employed the Carleman estimates with (68) and proved the Hölder stability estimates for some inverse source problems. If we assume suitable boundary conditions, for example, u(x, t) = 0, H(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), in additional to (1), we can easily combine the arguments in Choulli, Imanuvilov, Puel and Yamamoto [8] or Imanuvilov and Yamamoto [19], in which the authors discussed similar inverse source problems for the linearized Naiver-Stokes equations by the Carleman estimates with (69), with what we have done in this article to derive the global Lipschitz stability estimates for the inverse source problems. ...

... Recently in Imanuvilov and Yamamoto [10], the authors proved a conditional Lipschitz stability estimate as well as the uniqueness for the case t 0 = T . See also Huang, Imanuvilov and Yamamoto [7]. ...

... When Σ is an arbitrary nonempty open subset of Γ, a logarithmic stability inequality was recently established in [16] in the two dimensional case. ...

... For the nonlinear Oskolkov's system, the inverse problem with unknown source function is studied in [13]. Carleman estimates and its application of proving the Lipschitz stability of an inverse problem for the Kelvin-Voigt model is described in [14]. Taking the similar path as in [4], the hyperbolic type of inverse problems have been investigated by several authors to obtain global in time existence and uniqueness results (see [1,17,25], etc). ...

... Starting from the originating publication [4], Carleman estimates have been actively used for proofs of global uniqueness and stability results for coefficient inverse problems. Since this paper is not a survey of publications devoted to the method of [4], we refer now only to a few of those and references cited therein [3,6,9,10,11,12,13,14,15,18,40]. In addition, the idea of [4] was extended to numerical methods for coefficient inverse problems, see, e.g. ...

... Partial data results are also known for a two-dimensional Maxwell system [IY14a] and for Maxwell equations in a waveguide [IY14b], based on the two-dimensional partial data results of [IUY10], and in the case where the parameters are known near the boundary [BMR14]. Recently, extensions of the Carleman estimate and reflection approaches for partial data problems were introduced in [KS13] and [IY13]; some of these extensions for the Maxwell system were considered in [IY14c]. We also remark that the methods in the recent paper [DKLS13] might allow to relax to some extent the admissibility assumption in Theorem 1.1. ...

... Due to the partial symmetry of the domain one can reflect across the flat part of the boundary and reduce the problem to a full data problem. Partial data results are also known for a two-dimensional Maxwell system [IY14a] and for Maxwell equations in a waveguide [IY14b], based on the two-dimensional partial data results of [IUY10], and in the case where the parameters are known near the boundary [BMR14]. Recently, extensions of the Carleman estimate and reflection approaches for partial data problems were introduced in [KS13] and [IY13]; some of these extensions for the Maxwell system were considered in [IY14c]. ...

... The unique continuation problem for the Stokes equations was initially studied in [25]. The analysis of the stability properties of ill-posed problems based on the Navier-Stokes equations is a very active field of research, and we refer to the works [4,5,6,9,28,29,34] for recent results. ...

... This subsection is devoted to show an auxiliary optimal control problem which is useful in the proof of our main results, see [13], [14] and [15]. ...

... Later on, the fundamental work by Sylvester and Uhlmann proved the global uniqueness for the inverse potential problem when Γ D = Γ N = ∂Ω by constructing almost complex exponential solutions, which also yields the global uniqueness of the inverse conductivity problem [38]. Important progress on the partial data problem has been achieved in [7] and [3], [20], [21], [22], [23], [25], [30], [32], while there are many cases still remaining quite open, see [31] for a review. Up to now, there is no result for both Γ D and Γ N taken on arbitrary open subsets of the boundary except for the case when Γ D = Γ N (see e.g., [3] and [14]), to the best of our knowledge. ...