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 $\Omega\subset\mathbb{R}^{3}$, 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 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

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

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 $C^{1,1}$ 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

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