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Register a new userLorentzian Calder{o}n problem under curvature bounds
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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.
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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.
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 this paper, we establish compactness and existence results to a Branson-Paneitz type problem on a bounded domain of R^n with Navier boundary condition.
We consider the Calder`on problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
In this note we present a new proof of Sobolevs inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut-V{e}ron and V{e}rons 1991 classical work.
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