Harish Shrivastava
We study a free transmission problem in which solution minimizes a functional with different definitions in positive and negative phase of function. We prove some asymptotic regularity results when the jumps of the diffusion coefficients gets smaller along the free boundary. At last, we see a measure theoretic result related to the free boundary.
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We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solution like reflection, transmission and mode conversion of S and P waves, evanescent modes, Rayleigh and Stoneley waves. In particular, we recover Knott's equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to th…
Energy decay rate of a transmission system governed by degenerate wave equation with drift and under heat conduction with memory effect
本文研究退化波动方程与带记忆效应热方程耦合的传输系统镇定问题，证明在Coleman-Gurtin热传导下系统具有$t^{-4}$型多项式衰减率，在Gurtin-Pipkin热传导下具有指数稳定性。
We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.
We prove existence and regularity results for free transmission problems governed by fully nonlinear elliptic equations with nonhomogeneous degeneracies.
We obtain parabolic transmission eigenvalue-free regions for both isotropic andanisotropic interior transmission problems with coefficients which are Lipschitznear the boundary
Direct and inverse scattering for an isotropic medium with a second-order boundary condition
研究针对具有二级Robin边界条件的各向同性散射体，提出直接采样方法解决反向形状问题，并探讨传输特征值问题。
This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in \cite{CCH}. Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the m…
The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction $\sqrt{n(x)}$ is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that $n(x)-1$ does not vanish at the boundar…
Well-posedness for a transmission problem connecting first and second-order operators
本文研究在不相交区域上分别由椭圆型方程和程函型方程支配的传输问题，证明其粘性解的存在唯一性，动机来自随机最优控制中的 Hamilton-Jacobi 方程。
We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenome…
Fully nonlinear parabolic fixed transmission problems
针对时变界面两侧由不同完全非线性算子控制的抛物传输问题，证明 $C^{1,α}$ 界面下粘性解的分片 $C^{1,α}$ 正则性，并给出 ABP-Krylov-Tso 估计及平界面问题的适定性与正则性结论。
We study an equation governed by a discontinuous fully nonlinear operator. Such discontinuities are solution-dependent, which introduces a free boundary. Working under natural assumptions, we prove the existence of $L^p$-viscosity and strong solutions to the problem. The operator does not satisfy the usual structure conditions and to obtain the existence of solutions we resort to solving approximate problems, combined with an iterative scheme. We believe our strategy can be applied to other classes of non-variational free boundary problems.
In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue wi…
We study microlocally the transmission problem at the interface between an isotropic linear elastic solid and a linear inviscid fluid. We set up a system of evolution equations describing the particle displacement and velocity in the solid, and pressure and velocity in the fluid, coupled by suitable transmission conditions at the interface. We show well posedness for the coupled system and study the problem microlocally, constructing a parametrix for it using geometric optics. This construction describes the reflected and transmitted waves, including mode converted ones, related to incoming wa…
For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and Hörmander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotr…
Transmission Eigenvalues and Non-scattering
本文研究了线性霍姆霍兹方程中非散射现象的传输特征值问题，探讨了不同介质条件下非散射发生的条件及影响因素。
A parabolic free transmission problem: flat free boundaries are smooth
该研究针对复合材料相变电导率跳变引发的两相抛物型自由边界问题，通过线性化、紧性论证、Hodograph变换等方法证明平坦自由边界的高阶正则性，还提出新Harnack型不等式并发展了对应Schauder理论。
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the c…
In this paper, we establish the discreteness of transmission eigenvalues for Maxwell's equations. More precisely, we show that the spectrum of the transmission eigenvalue problem is discrete, if the electromagnetic parameters $\eps, \, μ, \, \heps, \, \hmu$ in the equations characterizing the inhomogeneity and background, are smooth in some neighborhood of the boundary, isotropic on the boundary, and satisfy the conditions $\eps \neq \heps$, $μ\neq \hmu$, and $\eps/ μ\neq \heps/ \hmu$ on the boundary. These are quite general assumptions on the coefficients which are easy to check. To our knowl…
We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.
This paper is concerned with the linear theory of thermoelasticity with microtemperatures, based on the entropy balance proposed by Green and Naghdi, which permits the transmission of heat as thermal waves of finite speed. We analyze the behavior of Rayleigh waves in an unbounded isotropic homogeneous strongly elliptic thermoelastic material with microtemperatures. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the elements of the bases of the kernels of appropriate matrices. The secular equation is established and afterwards an explicit form …
In this paper, we present a Spectral-Galerkin Method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl's law.…
Stochastic scattering control of spider diffusion governed by an optimal diffraction probability measure selected from its own local-time
本文研究Walsh蜘蛛扩散的随机最优散射控制问题，通过证明弱动态规划原理并利用HJB系统与非线性局部时间Kirchhoff传输边界条件唯一刻画值函数，基于比较定理解锁了此类问题的研究。
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