504 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
On some geometric properties of currents and Frobenius theorem
In this note we announce some results, due to appear in [2], [3], on the
structure of integral and normal currents, and their relation to Frobenius
theorem. In particular we show that an integral current cannot be tangent to a
distribution of planes which is nowhere involutive (Theorem 3.6), and that a
normal current which is tangent to an involutive distribution of planes can be
locally foliated in terms of integral currents (Theorem 4.3). This statement
gives a partial answer to a question raised by Frank Morgan in [1].Comment: 8 page
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
Exponential self-similar mixing and loss of regularity for continuity equations
We consider the mixing behaviour of the solutions of the continuity equation
associated with a divergence-free velocity field. In this announcement we
sketch two explicit examples of exponential decay of the mixing scale of the
solution, in case of Sobolev velocity fields, thus showing the optimality of
known lower bounds. We also describe how to use such examples to construct
solutions to the continuity equation with Sobolev but non-Lipschitz velocity
field exhibiting instantaneous loss of any fractional Sobolev regularity.Comment: 8 pages, 3 figures, statement of Theorem 11 slightly revise
- âŠ