Pedro Caro, Petri Ola, Mikko Salo
We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the bound- ary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov for the Schrödinger equation to Maxwell equations.
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An inverse boundary value problem for the Maxwell's equations with partial data
本文研究动态麦克斯韦方程组的逆边界问题，在存在严格凸叶状结构的条件下，证明了介电常数、电导率和磁导率的局部唯一确定性。
We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of R^3, from a finite number of boundary measurements. Our main result is a Hölder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell's system with boundary conditions.
In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.
We consider an inverse boundary value problem for Maxwell's equations, which aims to recover the electromagnetic material properties of a body from measurements on the boundary. We show that a Lipschitz continuous conductivity, electric permittivity, and magnetic permeability are uniquely determined by knowledge of all tangential electric and magnetic fields on the boundary of the body at a fixed frequency.
In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
This work is concerned with an inverse electromagnetic scattering problem in two dimensions. We prove that in the TE polarization case, the knowledge of the electric far-field pattern incited by a single incoming wave is sufficient to uniquely determine the shape of a penetrable scatterer of rectangular type. As a by-product, the uniqueness is also confirmed to inverse transmission problems modelled by scalar Helmholtz equations with discontinuous normal derivatives at the scattering interface. Keywords: Uniqueness, inverse medium scattering, Maxwell equations, one incoming wave, shape identif…
In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.
Inverse random source problem for Maxwell equations in an inhomogeneous medium
针对非均匀介质中白噪声驱动的随机 Maxwell 方程，本文证明正源问题适定性，并仅利用固定频率随机 Dirichlet 数据得到随机源强度的对数稳定性估计。
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
This paper concerns the quantitative step of the medical imaging modality Thermo-acoustic Tomography (TAT). We model the radiation propagation by a system of Maxwell's equations. We show that the index of refraction of light and the absorption coefficient (conductivity) can be uniquely and stably reconstructed from a sufficiently large number of TAT measurements. Our method is based on verifying that the linearization of the inverse problem forms a redundant elliptic system of equations. We also observe that the reconstructions are qualitatively quite different from the setting where radiation…
Pride (1994, Phys. Rev. B 50 15678-96) derived the governing model of electroseismic conversion, in which Maxwell's equations are coupled with Biot's equations through an electrokinetic mobility parameter. The inverse problem of electroseismic conversion was first studied by Chen and Yang (2013, Inverse Problem 29 115006). By following the construction of Complex Geometrical Optics (CGO) solutions to a matrix \Schrodinger equation introduced by Ola and Somersalo (1996, SIAM J. Appl. Math. 56 No. 4 1129-1145), we analyze the reconstruction of conductivity, permittivity and the electrokinetic mo…
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel ti…
This paper considers a class of nonlinear time harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form $\mathscr{X}(x,|\vec E(x)|^2)\vec E(x)$, $\mathscr{Y}(x,|\vec H(x)|^2)\vec H(x)$, such that $\mathscr{X}(x,s)$, $\mathscr{Y}(x,s)$ are both real analytic in $s$. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonl…
We consider for the full time-dependent Maxwell's equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements on the boundary for a finite time interval.
Inverse problem for the discrete Maxwell equations in a bounded paving
该研究针对有界铺砌中离散各向异性麦克斯韦方程组的逆问题，证明在Dirichlet-to-Neumann算子有唯一解时，可通过重构过程由该算子确定包含介质各向异性信息的对角算子$D_a$。
We consider a linearised inverse conductivity problem for electromagnetic waves in a three dimensional bounded domain at a high time-harmonic frequency. Increasing stability bounds for the conductivity coefficient in the full Maxwell system and in a simplified transverse electric mode are derived. These bounds contain a Lipschitz term with a factor growing polynomially in terms of the frequency, a Holder term, and a logarithmic term which decays with respect to the frequency as a power. To validate this increasing stability numerically, we propose a reconstruction algorithm aiming at the recov…
This paper is concerned with an inverse random source problem for the three-dimensional time-harmonic Maxwell equations. The source is assumed to be a centered complex-valued Gaussian vector field with correlated components, and its covariance operator is a pseudo-differential operator. The well-posedness of the direct source scattering problem is established and the regularity of the electromagnetic field is given. For the inverse source scattering problem, the micro-correlation strength matrix of the covariance operator is shown to be uniquely determined by the high frequency limit of the ex…
We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be $0, \infty$ or a finite non-identically vanishing variable function. We establish an accurate relationship between the vanishing order of the solutions to the Maxwell system and the dihedral angle of the edge-corner. In particular, if the angle is irrational, the vanishing order is infinity, i.e. strong unique continuation holds from the edge-corner. The establishment of those new qua…
The Boltzmann equation with non-isothermal Maxwell boundary conditions
研究非等温Maxwell边界条件下Boltzmann方程在接近水动力极限区域的解的存在性、唯一性及非平衡稳态的渐进行为。
We study the Landau-Lifshitz system associated with Maxwell equations in a bilayered ferromagnetic body when super-exchange and surface anisotropy interactions are present in the spacer in-between the layers. In the presence of these surface energies, the Neumann boundary condition becomes nonlinear. We prove, in three dimensions, the existence of global weak solutions to the Landau-Lifshitz-Maxwell system with nonlinear Neumann boundary conditions.
This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the electromagnetic field measured on a plane surface above the obstacle. For the direct problem, a new boundary integral equation is proposed and its well-posedness is establis…
In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in $H^m$. Moreover, we show the existence of a unique $H^m$-solution if the coefficients and the data are accordingly regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence condit…
On the local Maxwellians solving the Boltzmann equation with boundary condition
本文研究带边界 Boltzmann 方程的局部 Maxwell 分布，证明 bounce-back 条件下仅有全局解，并对二维、三维镜面反射条件下的区域及对称局部解进行完整分类。
This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to uniform bounds on solutions to the scattering problems for an extremely general class of admissible scatterers and inhomogeneities. The uniform bounds are a key step to tackle th…
In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m \geq 3$. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumpti…
We consider an initial-boundary value problem for the Maxwell's system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver-Muller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.
Maxwell's equations with mixed impedance boundary conditions
本文研究有界Lipschitz域上带矩阵值阻抗边界条件的时谐麦克斯韦方程组，推导Fredholm二择一定理，证明非共振频率下任意源的弱解存在性，并涵盖奇异阻抗系数情况。
We prove global stability results of {\sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the ki…
We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In …
We study Maxwell's equations in time domain in an anisotropic medium. The goal of the paper is to solve an inverse boundary value problem for anisotropies characterized by scalar impedance $α$. This means that the material is conformal, i.e., the electric permittivity $ε$ and magnetic permeability $μ$ are tensors satisfying $μ=α^2ε$. This condition is equivalent to a single propagation speed of waves with different polarizations which uniquely defines an underlying Riemannian structure. The analysis is based on an invariant formulation of the system of electrodynamics as a Dirac type first ord…
Starting from the local-in-time classical solution to the compressible Euler system with impermeable boundary condition in half-space, by employing the coupled weak viscous layers (governed by linearized compressible Prandtl equations with Robin boundary condition) and linear kinetic boundary layers, and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new boundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time existence of Hilbert expansion type classical solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition with acco…
In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m \geq 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m \geq 3$ of the corresponding initial boundary value problem if the material laws and the data are accor…
Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions
本文研究理想导体边界下二维、三维 Maxwell 方程，通过边界延拓与半波对角化恢复至端点的 Strichartz 估计，并改进二维 Kerr 非线性 Maxwell 系统的局部适定性。
In this note we consider boundary value problems in electromagnetism. We prove well-posedness results for the time-harmonic Maxwell equations in the setting of Riemannian manifolds. We also consider the eigenvalue problem the homogeneous time-harmonic Maxwell equations with zero boundary conditions.
We consider relativistic plasma particles subjected to an external gravitation force in a $3$D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov-Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. We construct weak solutions to these initial-boundary v…
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.
Exponential decay of the linear Maxwell system due to conductivity near the boundary
该研究证明了在边界附近存在导电性时，线性Maxwell系统解会指数衰减，通过霍尔兹分解和观测型估计方法实现。
We are interested in the homogenization of energy like quantities for electromagnetic waves in the high frequency limit for Maxwell's equations with various boundary conditions. We use a scaled variant of H-measures known as semi classical measures or Wigner measures. Firstly, we consider this system in the half space of $\R^3$ in the time harmonic and with conductor boundary condition at the flat boundary $x_3=0$. Secondly we consider the same system but with Calderon boundary condition. Thirdly, we consider this system in the curved interface case.
When a porous rock is saturated with an electrolyte, electrical fields are coupled with seismic waves via the electro-seismic conversion. Pride derived the governing models, in which Maxwell equations are coupled with Biot equations through the electro-kinetic mobility parameter. The inverse problem of the linearized electro-seismic conversion consists in two step, namely the inversion of Biot equations and the inversion of Maxwell equations. We analyze the reconstruction of conductivity and electro-kinetic mobility parameter in Maxwell equations with internal measurements, while the internal …
We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & Müller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is …
Kinetic Fokker-Planck equations with Maxwell boundary conditions
研究具有麦克斯韦边界条件的线性动力学福克 - 普朗克方程解的边界正则性理论，通过调节系数\(\alpha\in(0,1)\)解决中间区域问题，证明解的赫尔德连续性及建立特定阶边界正则性，还发展了可扩展的统一方法。
We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response opera…
Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary
本文研究了具有吸收边界条件的非线性Maxwell系统，证明了小数据解的全局存在性和指数衰减性，并通过迹估计和正则性分析改进了线性自治情况下的结果。
Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition
该研究针对带硬球碰撞和一般Maxwell反射边界条件的Boltzmann方程，构造含内部流体与Knudsen-粘性耦合边界层的解，证明了适应系数在$0<α≤1$全范围的声学极限，严格验证了Sone的形式分析，填补了经典解框架下非镜面反射情形的研究空白。
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modi…
The Boltzmann equation on smooth and cylindrical domains with Maxwell boundary conditions
本文研究有界域上Boltzmann方程在流体动力学极限附近的适定性，针对两类边界条件（Maxwell和圆柱域混合反射）使用多种权重构建定量Cauchy理论。
Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits
本文研究半空间Knudsen层方程，针对带角截断硬球碰撞与调节系数0<α<1的Maxwell反射边界条件，证明解的存在唯一性与渐近性，并应用于流体极限推导流体方程边界条件，刻画了消失源集。
The Local-well-posedness of the relativistic Vlasov-Maxwell-Landau system with the specular reflection boundary condition
本文证明相对论 Vlasov-Maxwell-Landau 系统在含镜面反射边界的三维有界（含非凸）区域中的时间局部适定性，首次涵盖自洽磁场效应。
Shape Derivatives for Maxwell's Equations with Nonlinear Boundary Conditions
本文针对具有点态非线性边界和界面响应的时间谐波麦克斯韦散射问题，建立了迹正则变分框架，并证明了形状导数的Hadamard结构。
In bounded convex domains, the regularity estimates of a vector field $\u$ with its $\dv\u$, $\curl\u$ in $L^r$ space and the tangential components or the normal component of $\u$ over the boundary in $L^r$ space, are established for $1<r<\infty$. As an application, we derive an $H^r(\curl,Ø)$ estimate for solutions to a Maxwell-type system with an inhomogeneous boundary condition in convex domains.
We establish the global-in-time existence and uniqueness of classical solutions to the "one and one-half" dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studie…
Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition
本文在Maxwell边界条件下，通过强解框架建立了Boltzmann方程到Navier-Stokes-Fourier系统的强扩散极限，并引入了ε拉伸方法和旋转Maxwellian的耗散估计作为主要创新。
Increasing stability for the inverse source problems in electrodynamics
本文研究时间依赖 Maxwell 方程逆源问题的增强稳定性，利用 Fourier 变换、解析延拓锐界和 Huygens 原理，获得了源函数 L^2 范数的增强稳定性估计。
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.
The goal of this work is to study the electromagnetic scattering problem of time-domain Maxwell's equations in an unbounded structure. An exact transparent boundary condition is developed to reformulate the scattering problem into an initial-boundary value problem in an infinite rectangular slab. The well-posedness and stability are established for the reduced problem. Our proof is based on the method of energy, the Lax--Milgram lemma, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the electric field by dir…
Incompressible Euler limit from the Boltzmann equation with Maxwell reflection boundary condition in the half-space
本文严格证明了半空间中具有麦克斯韦反射边界条件的玻尔兹曼方程在剪切流情形下的不可压缩欧拉极限，通过守恒律转化法向导数损失并利用解析框架建立一致估计。
Linear stability of traveling Maxwellians for Landau equation with very soft potentials
本文研究具极软势的朗道方程行进麦克斯韦分布的线性稳定性，证明其线性解微观部分逆多项式衰减、宏观部分有界不衰减，且其长时间动力学与两类对比情况存在差异。
In this paper, we consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. The scattering interface is supposed to be the graph of a Lipschitz continuous function with compact support. It is proved that an acoustically sound-soft or sound-hard surface can be uniquely determined by the far-field pattern of infinite number of incident plane waves with distinct directions. Moreover, a single point source or plane wave can be used to uniquely determine a scattering surface of polyhedral type. These uniqueness results apply to…
We study inverse problems for the Einstein-Maxwell equations. We prove that it is possible to generate gravitational waves from the nonlinear interactions of electromagnetic waves. By sending electromagnetic waves from a neighborhood of a freely falling observer and taking measurements of the gravitational perturbations in the same neighborhood, one can determine the vacuum space-time structure up to diffeomorphisms in the largest region where these waves can travel to from the observer and return.
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