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

On multiple frequency power density measurements II. The full Maxwell's equations

We shall give conditions on the illuminations $\varphi_{i}$ such that the solutions to Maxwell's equations $$\left\{ \begin{array}{l} {\rm curl} E^{i}=i\omega\mu H^{i}\qquad\text{in }\Omega,\\ {\rm curl} H^{i}=-i(\omega\varepsilon+i\sigma)E^{i}\qquad\text{in }\Omega,\\ E^{i}\times\nu=\varphi_{i}\times\nu\qquad\text{on }\partial\Omega, \end{array}\right.$$ satisfy certain non-zero qualitative properties inside the domain $\Omega$, provided that a finite number of frequencies $\omega$ 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

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 $\Omega\subset\mathbb{R}^{3}$, given by $$\left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_{\omega}^{g})-\omega qu_{\omega}^{g}=0\quad\text{in $\Omega$,}\\ u_{\omega}^{g}=g\quad\text{on $\partial\Omega$.} \end{array}\right.$$ We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overline{\Omega}=\cup_{\omega\in K}\Omega_{\omega}$ such that $|\nabla u_{\omega}^{g}(x)|>0$ for every $\omega\in K$ and $x\in\Omega_{\omega}$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $\Omega$, the admissibility condition on $g$ is a generic property.Comment: 14 page

On Multiple Frequency Power Density Measurements

We shall give a priori conditions on the illuminations $\phi_i$ such that the solutions to the Helmholtz equation $-div(a \nabla u^i)-k q u^i=0$ in \Omega, $u^i=\phi_i$ on $\partial\Omega$, 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

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

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 $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+\delta}$ for some $\delta>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ 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

Novel diagnostic for precise measurement of the modulation frequency of Seeded Self-Modulation via Coherent Transition Radiation in AWAKE

We present the set-up and test-measurements of a waveguide-integrated heterodyne diagnostic for coherent transition radiation (CTR) in the AWAKE experiment. The goal of the proof-of-principle experiment AWAKE is to accelerate a witness electron bunch in the plasma wakefield of a long proton bunch that is transformed by Seeded Self-Modulation (SSM) into a train of proton micro-bunches. The CTR pulse of the self-modulated proton bunch is expected to have a frequency in the range of 90-300 GHz and a duration of 300-700 ps. The diagnostic set-up, which is designed to precisely measure the frequency and shape of this CTR-pulse, consists of two waveguide-integrated receivers that are able to measure simultaneously. They cover a significant fraction of the available plasma frequencies: the bandwidth 90-140 GHz as well as the bandwidth 255-270 GHz or 170-260 GHz in an earlier or a latter version of the set-up, respectively. The two mixers convert the CTR into a signal in the range of 5-20 GHz that is measured on a fast oscilloscope, with a high spectral resolution of 1-3 GHz dominated by the pulse length. In this contribution, we will describe the measurement principle, the experimental set-up and a benchmarking of the diagnostic in AWAKE.Comment: Conference proceedings to 3rd European Advanced Accelerator Concepts Worksho

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 $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $\sigma$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $\sigma(x)$ such that $\nabla u$ vanishes at least at one point in $X$. 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 $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $\sigma(x)$. 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 $\sigma(x)$ 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