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Mikko Salo

Publications and source records attributed to Mikko Salo.

108 records · Page 6Linked to original sources
arXiv2012

The attenuated ray transform for connections and Higgs fields

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.

math.DG↗
arXiv2011

On $L^p$ resolvent estimates for Laplace-Beltrami operators on compact manifolds

In this article we prove $L^p$ estimates for resolvents of Laplace-Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge in the Euclidean case and Shen for the torus. We follow Sogge and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain $L^p$ Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig; we illustrate the pertinence of $L^p$ resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the construction of complex geometrical optics solutions to the Schrödinger equation with unbounded potentials, an essential device for solving anisotropic inverse problems.

math.AP↗
arXiv2011

An inverse problem for the p-Laplacian: boundary determination

We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.

math.AP↗
arXiv2011

Determining an unbounded potential from Cauchy data in admissible geometries

In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n \geq 3. In this article we extend this result to the case of unbounded potentials, namely those in Ln/2. In the process, we derive Lp Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison-Kenig and Kenig-Ruiz-Sogge.

math.AP↗
arXiv2010

Nowhere conformally homogeneous manifolds and limiting Carleman weights

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analog of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.

math.DG↗
arXiv2010

Reconstructions from boundary measurements on admissible manifolds

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-Δ_g + q$ in a fixed admissible 3-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $Δ_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. on admissible manifolds, and extends the reconstruction procedure of Nachman in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.

math.AP↗
arXiv2010

Inverse scattering at fixed energy on surfaces with Euclidean ends

On a fixed Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we show that, under a topological condition, the scattering matrix $S_V(\la)$ at frequency $\la > 0$ for the operator $Δ+V$ determines the potential $V$ if $V\in C^{1,α}(M_0)\cap e^{-γd(\cdot,z_0)^j}L^\infty(M_0)$ for all $γ>0$ and for some $j\in\{1,2\}$, where $d(z,z_0)$ denotes the distance from $z$ to a fixed point $z_0\in M_0$. The topological condition is given by $N\geq\max(2g+1,2)$ for $j=1$ and by $N\geq g+1$ if $j=2$. In $\rr^2$ this implies that the operator $S_V(\la)$ determines any $C^{1,α}$ potential $V$ such that $V(z)=O(e^{-γ|z|^2})$ for all $γ>0$.

math.AP↗
arXiv2009

Inverse scattering for the magnetic Schroedinger operator

We show that fixed energy scattering measurements for the magnetic Schroedinger operator uniquely determine the magnetic field and electric potential in dimensions $n \geq 3$. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schroedinger equation via a pseudodifferential conjugation argument.

math.AP↗
arXiv2009

Inverse problems for the anisotropic Maxwell equations

We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.

math.AP↗
arXiv2009

Inverse boundary value problem for Maxwell equations with local data

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the bound- ary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov for the Schrödinger equation to Maxwell equations.

math.AP↗
arXiv2009

Inverse problems with partial data for a Dirac system: a Carleman estimate approach

We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.

math.AP↗
arXiv2008

Carleman estimates and inverse problems for Dirac operators

We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.

math.AP↗
arXiv2006

Stability for solutions of wave equations with C^{1,1} coefficients

We consider the stable dependence of solutions to wave equations on metrics in C^{1,1} class. The main result states that solutions depend uniformly continuously on the metric, when the Cauchy data is given in a range of Sobolev spaces. The proof is constructive and uses the wave packet approach to hyperbolic equations.

math.AP↗
arXiv2006

Determining nonsmooth first order terms from partial boundary measurements

We extend results of Dos Santos Ferreira-Kenig-Sjoestrand-Uhlmann (math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schroedinger operator determine uniquely the magnetic field related to a Hoelder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.

math.AP↗
arXiv2006

Complex spherical waves and inverse problems in unbounded domains

We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman estimate. Using these solutions we can treat the inverse problem in the unbounded domain like in the bounded case, and it is possible to localize the boundary measurements to a finite part of the boundary.

math.AP↗