121 research outputs found
H\"older regularity for Maxwell's equations under minimal assumptions on the coefficients
We prove global H\"older regularity for the solutions to the time-harmonic
anisotropic Maxwell's equations, under the assumptions of H\"older continuous
coefficients. The regularity hypotheses on the coefficients are minimal. The
same estimates hold also in the case of bianisotropic material parameters.Comment: 11 page
Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems
We study the boundary control of solutions of the Helmholtz and Maxwell
equations to enforce local non-zero constraints. These constraints may
represent the local absence of nodal or critical points, or that certain
functionals depending on the solutions of the PDE do not vanish locally inside
the domain. Suitable boundary conditions are classically determined by using
complex geometric optics solutions. This work focuses on an alternative
approach to this issue based on the use of multiple frequencies. Simple
boundary conditions and a finite number of frequencies are explicitly
constructed independently of the coefficients of the PDE so that the
corresponding solutions satisfy the required constraints. This theory finds
applications in several hybrid imaging modalities: some examples are discussed.Comment: 24 pages, 2 figure
Absence of Critical Points of Solutions to the Helmholtz Equation in 3D
The focus of this paper is to show the absence of critical points for the
solutions to the Helmholtz equation in a bounded domain
, given by We prove that for an admissible there exists a finite
set of frequencies in a given interval and an open cover
such that for every and . The
set is explicitly constructed. If the spectrum of the above problem is
simple, which is true for a generic domain , the admissibility
condition on is a generic property.Comment: 14 page
On multiple frequency power density measurements II. The full Maxwell's equations
We shall give conditions on the illuminations such that the
solutions to Maxwell's equations satisfy certain non-zero qualitative properties inside
the domain , provided that a finite number of frequencies are
chosen in a fixed range. The illuminations are explicitly constructed. This
theory finds applications in several hybrid imaging problems, where unknown
parameters have to be imaged from internal measurements. Some of these examples
are discussed. This paper naturally extends a previous work of the author
[Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied.Comment: 24 page
On Multiple Frequency Power Density Measurements
We shall give a priori conditions on the illuminations such that the
solutions to the Helmholtz equation in \Omega,
on , and their gradients satisfy certain non-zero
and linear independence properties inside the domain \Omega, provided that a
finite number of frequencies k are chosen in a fixed range. These conditions
are independent of the coefficients, in contrast to the illuminations
classically constructed by means of complex geometric optics solutions. This
theory finds applications in several hybrid problems, where unknown parameters
have to be imaged from internal power density measurements. As an example, we
discuss the microwave imaging by ultrasound deformation technique, for which we
prove new reconstruction formulae.Comment: 26 pages, 4 figure
Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coefficients
The focus of this paper is the study of the regularity properties of the time
harmonic Maxwell's equations with anisotropic complex coefficients, in a
bounded domain with boundary. We assume that at least one of the
material parameters is for some . Using regularity
theory for second order elliptic partial differential equations, we derive
estimates and H\"older estimates for electric and magnetic fields up
to the boundary. We also derive interior estimates in bi-anisotropic media.Comment: 19 page
Critical Points for Elliptic Equations with Prescribed Boundary Conditions
This paper concerns the existence of critical points for solutions to second
order elliptic equations of the form posed on
a bounded domain with prescribed boundary conditions. In spatial dimension
, it is known that the number of critical points (where ) is
related to the number of oscillations of the boundary condition independently
of the (positive) coefficient . We show that the situation is different
in dimension . More precisely, we obtain that for any fixed (Dirichlet
or Neumann) boundary condition for on , there exists an open
set of smooth coefficients such that vanishes at least
at one point in . By using estimates related to the Laplacian with mixed
boundary conditions, the result is first obtained for a piecewise constant
conductivity with infinite contrast, a problem of independent interest. A
second step shows that the topology of the vector field on a
subdomain is not modified for appropriate bounded, sufficiently high-contrast,
smooth coefficients .
These results find applications in the class of hybrid inverse problems,
where optimal stability estimates for parameter reconstruction are obtained in
the absence of critical points. Our results show that for any (finite number
of) prescribed boundary conditions, there are coefficients for
which the stability of the reconstructions will inevitably degrade.Comment: 26 pages, 4 figure
Mathematical Analysis of Ultrafast Ultrasound Imaging
This paper provides a mathematical analysis of ultrafast ultrasound imaging.
This newly emerging modality for biomedical imaging uses plane waves instead of
focused waves in order to achieve very high frame rates. We derive the point
spread function of the system in the Born approximation for wave propagation
and study its properties. We consider dynamic data for blood flow imaging, and
introduce a suitable random model for blood cells. We show that a singular
value decomposition method can successfully remove the clutter signal by using
the different spatial coherence of tissue and blood signals, thereby providing
high-resolution images of blood vessels, even in cases when the clutter and
blood speeds are comparable in magnitude. Several numerical simulations are
presented to illustrate and validate the approach.Comment: 25 pages, 13 figure
Calder\uf3n's inverse problem with a finite number of measurements II: independent data
We prove a local Lipschitz stability estimate for Gel'fand-Calder\uf3n's inverse problem for the Schr\uf6dinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of L 1e. A similar result for Calder\uf3n's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors
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