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Register a new userThe Calder{o}n inverse problem for isotropic quasilinear conductivities
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We prove a global uniqueness result for the Calder{o}n inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension $nge 3$. Performing higher order linearizations of the nonlinear Dirichlet--to--Neumann map, we reduce the problem of the recovery of the differentials of the quasilinear conductivity, which are symmetric tensors, to a completeness property for certain anisotropic products of solutions to the linearized equation. The completeness property is established using complex geometric optics solutions to the linearized conductivity equation, whose amplitudes concentrate near suitable two dimensional planes.
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We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderon problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.
The aim of this paper is to establish global Calder{o}n--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -operatorname{div}(| abla u|^{p-2} abla u) = operatorname{div} (|F|^{p-2}F)~text{in}~Omegatimes (0,T)subset mathbb{R}^{n+1}, $$ proving that $$Fin L^qRightarrow abla uin L^q,$$ for any $q>max{p,frac{n(2-p)}{2}}$ and $p>1$. Acerbi and Mingione cite{Acerbi07} proved this estimate in the case $p>frac{2n}{n+2}$. In this article we settle the case $1<pleq frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.
In these notes we prove log-type stability for the Calderon problem with conductivities in $ C^{1,varepsilon}(bar{Omega}) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(bar{Omega}) $.
In this paper, we study an inverse coefficients problem for two coupled Schr{o}dinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations hold. A logarithmic type stability result is obtained. The main method is based on the Carleman estimate for coupled Schr{o}dinger equations and coupled heatn equations, and the Fourier-Bros-Iagolnitzer transform.
We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solution like reflection, transmission and mode conversion of S and P waves, evanescent modes, Rayleigh and Stoneley waves. In particular, we recover Knotts equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to them, under an additional condition of lack of full internal reflection of some of the waves.
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