Yernat M. Assylbekov
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.
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We establish the existence of semiclassical states for a nonlinear Klein-Gordon-Maxwell-Proca system in static form, with Proca mass 1 on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of 3-dimensional manifolds on which the system has semiclassical solutions for every exponent p>2. The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero
In this paper we are focusing to the following Schrödinger-Maxwell system $(\mathcal{SM}_{Ψ(λ,\cdot)}^{e})$: \[ \begin{cases} -Δ_{g}u+β(x)u+euϕ=Ψ(λ,x)f(u) & \mathrm{in}\ M -Δ_{g}ϕ+ϕ=qu^{2} & \mathrm{\mathrm{in}\ M} \end{cases} \] where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $β\in C^{\infty}(M)$ and $Ψ\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{Ψ(λ,\cdot)}^{e})$ …
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.
Regularity of solutions to time-harmonic Maxwell's system with Hölder and various lower than Hölder continuous coefficients
本文为二阶线性椭圆方程与时谐 Maxwell 系统建立完整 Schauder 理论，并在 Hölder 连续系数下证明时谐各向异性 Maxwell 方程解的全局 Hölder 正则性，将 Hölder 指数提升至 (0,1)。
Dual variational methods for time-harmonic nonlinear Maxwell's equations
本文利用对偶变分方法和无需 Palais-Smale 条件的新版对称山路定理，证明了有界域及 $\mathbb{R}^3$ 上时谐非线性 Maxwell 方程存在无穷多个非平凡解。
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.
We prove resolvent estimates in $L^p$-spaces for time-harmonic Maxwell's equations in two spatial dimensions and in three dimensions in the partially anisotropic case. In the two-dimensional case the estimates are sharp up to endpoints. We consider anisotropic permittivity and permeability, which are both taken to be time-independent and spatially homogeneous. For the proof we diagonalize time-harmonic Maxwell's equations to equations involving Half-Laplacians. We apply these estimates to infer a Limiting Absorption Principle in intersections of $L^p$-spaces and to localize eigenvalues for per…
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $Ω\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
In this work we investigate the L^p-L^q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an L^p-L^q-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.
On polarization interface conditions for time-harmonic Maxwell's equations
本文针对由细金属丝夹杂均匀化得到的极化界面条件，分析相应的时谐 Maxwell 极限方程，证明解的存在性并建立 Fredholm 二择一性。
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.
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…
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 $W^{1,3+δ}$ for some $δ>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and Hölder estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.
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 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…
On Landau equation with harmonic potential: nonlinear stability of time-periodic Maxwell-Boltzmann distributions
针对带调和势的Landau方程，严格证明Boltzmann的相关假设，证实熵不变解等价于时间周期Maxwell-Boltzmann分布且具有非线性稳定性，收敛速率由13条守恒律决定且达线性情形下的最优水平。
The Calderón operator for time-harmonic Maxwell's equations in closed waveguides
针对具有非均匀参数的闭合圆柱波导中的时谐麦克斯韦方程，证明了Calderón算子的有界性，分析了相关内部麦克斯韦问题的适定性，并给出了波导波数位置与群速度的额外结果。
Harmonic morphisms of sub-Riemannian Lie groups
本文证明次黎曼李群间的调和态射是光滑共形淹没且为拉普拉斯算子的对称性，给出李群外调和态射的部分结果，计算其水平能量第一变分，发现其不一定是调和映射。
We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R^m into a pseudo-Riemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point is actually smooth harmonic everywhere.
On the behaviour of harmonic functions on Riemannian cones
本文探讨了黎曼锥上调和函数的性质，并研究了Liouville定理在该情境下的成立情况。
Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media
本文研究了 $\mathbb{R}^3$ 中时谐 Maxwell 方程的解在满足单调性条件的 $L^\infty$ 系数下的先验界问题，提出了对任意大频率有效的显式界，并首次给出了可穿透 $C^0$ 星形障碍物散射问题的尖锐频率显式界。
We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability $μ$ and the permittivity $\varepsilon$ are symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We show that a unique continuation result for globally $W^{1,\infty}$ coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise $W^{1,\infty}$ with respect to a suitable countable collection of sub-domains with $C^{0}$ boundaries. Such suitable collections include any bounded finite collection. The p…
In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a new approach by giving $\mathcal{H}^1$ estimate when the coefficients are $\mathcal{L}^{\infty}$ bounded; and then we derive $\mathcal{W}^{1,p}$ estimates for every $p > 2$ when one of the leading coefficients is simply continuous; Finally, we extend the result to $\mathcal{C}^{1,α}$ almost everywhere for the solution of the homogeneous Maxwell's equations w…
We carry out the homogenization of time-harmonic Maxwell's equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale…
Here we derive some results on so called quantitative Runge approximation in the case of the time-harmonic Maxwell equations. This provides a Runge approximation having more explicit quantitative information. We additionally derive some results on the conditional stability of the Cauchy problem for the time-harmonic Maxwell equations.
We find solutions $E:Ω\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + λE = \partial_E F(x,E) &&\quad \text{in}Ω\\ &ν\times E = 0 &&\quad \text{on}\partialΩ\end{aligned} \right. \] on a simply connected, smooth, bounded domain $Ω\subset\mathbb{R}^3$ with connected boundary and exterior normal $ν:\partialΩ\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$, the nonlinearity $F:Ω\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=Γ(x)|E|^p$ for $Γ\in L^\inft…
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between complex Euclidean spaces.
Gradient estimates for $(p,V)$-harmonic functions on Riemannian manifolds
本文基于$(p,V)$-调和函数的梯度估计，利用Moser迭代方法在黎曼流形上建立了体积比较定理和Sobolev嵌入定理，并得到了显式全局梯度估计。
Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on …
On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.
In this paper we prove uniqueness for an inverse boundary value problem (IBVP) arising in electrodynamics. We assume that the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, are described by continuously differentiable functions.
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In this article we initiate a rigorous mathematical treatment of this functional and study various basic aspects of its critical points.
We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are al…
In the planar setting the Radó-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way …
We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic map.
We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replacements for Dirichlet-Neumann fields. Moreover, we even show convergence in operator norm.
In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the connection of this map to Herglotz functions for isotropic and anisotropic multicomponent composites.
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
We discuss the radiation problem of total reflection for a time-harmonic generalized Maxwell system in a non-smooth exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the identity. By means of the limiting absorption principle, a Fredholm alternative holds true and the eigensolutions decay polynomially resp. exponentially at infinity. We prove that the corresponding eigenvalues do not accumulate even at zero. Next, we show the convergence of the time-harmonic solutions to a solution of an electro-magneto static Maxwell s…
We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate in R \ {0} and that by means of Eidus' limiting absorption principle a Fredholm alternative holds true.
We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby, we identify degenerate operators in terms of special 'polynomially growing' solutions of a corresponding static problem, which must be added to the 'usual' Neumann series in order to describe the low frequency asymptotic adequately.
Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations, II
本文提出一种新的积分表示，用于求解R^3中分段常数介电常数和磁导率介质中的时谐Maxwell方程，导出第二类Fredholm积分方程组，避免虚假共振和低频崩溃，并通过数值算例展示性能。
Let $Ω\subset {R}^n,$ $n \geq 3,$ be a bounded open set, $x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $Ω,$ $u \colon Ω\to {R}^N ,$ $N>1,$ and $ Du=(D_αu^i)$, $D_α= \partial/\partial x_α, $ $α=1,\ldots,n,\,$ $i=1,\ldots,N .\,$ Main goal is the study of regularity of the minima of nondifferentiable functionals $$ {\cal F} \,=\, \int_ΩF(x,u,Du) dx. $$ having the integrand function different shapes of smoothness. The method is based on the use some majorizations for the functional, rather than the well known Euler equation associated to it.
In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savaré. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.
The Sesquiharmonic Map Flow from Riemannian Surfaces
研究从二维紧流形到紧流形的半调和映射流的L²梯度流，推广了Struwe关于调和映射的正则性结果。
$\varepsilon$-approximability and Quantitative Fatou Theorem on Riemannian Manifolds
本文将ε-逼近引理推广到完备黎曼流形，证明Lipschitz区域上的定量Fatou定理，并应用于调和函数及A-调和函数，推广了Dahlberg等人的工作。
Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations
本文证明了时谐麦克斯韦方程组Nédélec棱边有限元协调逼近的渐近最优性，并在最小正则性假设下建立了离散inf-sup稳定性，适用于一般区域、材料系数和右端项。
The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\nabla\times\left(μ(x)^{-1} \nabla\times u\right) - ω^2\varepsilon(x)u = f(x,u)$$ for the field $u:Ω\to\mathbb{R}^3$ in a domain $Ω\subset\mathbb{R}^3$. Here $\varepsilon(x) \in \mathbb{R}^{3\times3}$ is the (linear) permittivity tensor of the material, and $μ(x) \in \mathbb{R}^{3\times3}$ denotes the magnetic permeability tensor. The nonlinearity $f:Ω\times\mathbb{R}^3\to\mathbb{R}^3$ comes from the nonlinear polarization. If $f=\nabla_uF$ is a gradient…
Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations
本文通过发展一种新的出射解表示，导出第二类Fredholm积分方程，用于求解完美导体散射问题，避免了虚假共振和低频崩溃，并给出了非单连通区域上k-Neumann场存在性的新证明。
Limiting absorption principle and well-posedness for the time-harmonic Maxwell equations with anisotropic sign-changing coefficients
研究时谐域中具有各向异性变号系数的 Maxwell 方程，通过变量替换建立 Cauchy 问题并采用 Fourier 方法与变分方法推导先验估计，获得了具有局部特征且易于验证的系数条件，从而证明极限吸收原理与适定性，为负折射率超材料中电磁场稳定性提供充分判据。
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