Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
Wed, 20 Feb 2019 03:28:25 GMT2019-02-20T03:28:25ZUnique determination of the electric potential in the presence of a fixed magnetic potential in the plane
http://hdl.handle.net/20.500.11824/935
Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane
Caro P.; Rogers K.
For electric and magnetic potentials with compact support, we consider the magnetic Schrödinger equation with fixed positive energy. Under a mild additional regularity hypothesis, and with fixed magnetic potential, we show that the scattering solutions uniquely determine the electric potential. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.
Sat, 01 Dec 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9352018-12-01T00:00:00ZDetermination of convection terms and quasi-linearities appearing in diffusion equations
http://hdl.handle.net/20.500.11824/934
Determination of convection terms and quasi-linearities appearing in diffusion equations
Caro P.; Kian Y.
We consider the highly nonlinear and ill posed inverse problem of determining some general expression appearing in the a diffusion equation from measurements of solutions on the lateral boundary. We consider both linear and nonlinear expression. In the linear case, the equation is a convection-diffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by non-smooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasilinear expression appearing in a non-linear parabolic equation. Our result give a positive answer to the unique recovery of a general vector valued first order coefficient, depending on both time and space variable, and to the unique recovery inside the domain of quasilinear terms, from measurements restricted to the lateral boundary, for diffusion equations.
Sat, 01 Dec 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9342018-12-01T00:00:00ZCorrelation imaging in inverse scattering is tomography on probability distributions
http://hdl.handle.net/20.500.11824/933
Correlation imaging in inverse scattering is tomography on probability distributions
Caro P.; Helin T.; Kujanpää A.; Lassas M.
Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from empirical correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same correlations, they have identical laws as function-valued random variables.
Tue, 04 Dec 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9332018-12-04T00:00:00ZTwo-weight mixed norm estimates for a generalized spherical mean Radon transform acting on radial functions
http://hdl.handle.net/20.500.11824/926
Two-weight mixed norm estimates for a generalized spherical mean Radon transform acting on radial functions
Ciaurri Ó.; Nowak, A.; Roncal, L.
We investigate a generalized spherical means operator,
viz. generalized spherical mean Radon transform, acting on radial functions.
We establish an integral representation of this operator and find precise
estimates of the corresponding kernel.
As the main result, we prove two-weight mixed norm estimates for the integral operator, with
general power weights involved. This leads to weighted Strichartz type estimates for solutions
to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations with radial initial data
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9262018-01-01T00:00:00Z