63 research outputs found
Calderon inverse Problem with partial data on Riemann Surfaces
On a fixed smooth compact Riemann surface with boundary , we show
that for the Schr\"odinger operator with potential for some , the Dirichlet-to-Neumann map
measured on an open set determines
uniquely the potential . We also discuss briefly the corresponding
consequences for potential scattering at 0 frequency on Riemann surfaces with
asymptotically Euclidean or asymptotically hyperbolic ends.Comment: 27 pages. Corrections and modifications in the Complex Geometric
Optics solutions; regularity assumption strenghtened to $C^{1,\alpha}
Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations
This paper is a continuation of [13], where new variational principles were
introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue
here the program of using these Lagrangians to provide variational formulations
and resolutions to various basic equations and evolutions which do not normally
fit in the Euler-Lagrange framework. In particular, we consider stationary
equations of the form as well as i dissipative
evolutions of the form were is a convex potential on an infinite dimensional space. In
this paper, the emphasis is on the cases where the differential operators
involved are not necessarily bounded, hence completing the results established
in [13] for bounded linear operators. Our main applications deal with various
nonlinear boundary value problems and parabolic initial value equations
governed by the transport operator with or without a diffusion term.Comment: 30 pages. For the most updated version of this paper, please visit
http://www.pims.math.ca/~nassif/pims_papers.htm
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.Comment: 23 page
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