Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

Integral Equations with Modified Fundamental Solution in Time-Harmonic Electromagnetic Scattering

To overcome the non-uniqueness problem arising in integral equationsfor the exterior boundary-value problems for the scalar Helmholtzequation, Ursell and Jones suggested adding a series of outgoingwaves to the free-space fundamental solution. The present papergives an extension of this approach to the case of the reducedMaxwell equations of electromagnetic scattering. For the caseof a sphere an explicit choice of coefficients in the modificationis presented which is optimal with respect to various criteria.     

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

We prove that the standard second‐kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign‐definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so‐called combined potential, or combined field, operator.) This coercivity result yields k‐explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical‐asymptotic methods developed recently). Coercivity also gives k‐explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise‐polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation. © 2015 Wiley Periodicals, Inc.     

Estimates for the Approximation Numbers of One Class of Integral Operators. I

Asymptotic estimates are obtained for the approximation numbers of the Hardy integral operator with variable limits of integration.     

Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.     

Boundary Hypersingular Integral Delay Equation for Nonstationary Problems of Scattering by Perfectly Conducting Bodies

Differential Equations - We consider the application of the method of boundary integral equations to the problem of scattering of an electromagnetic field obeying the nonstationary Maxwell...     

电磁散射问题手性障碍边界重构的线性探测法的理论分析

In this paper,we consider the inverse scattering by chiral obstacle inelectromagnetic fields,and prove that the linear sampling method is also effective todetermine the support of a chiral obstacle from the noisy far field data.     

一类具有P-Laplacian算子的非线性分数阶微分方程解的存在性研究

通过构造Green函数的性质,利用Banach不动点定理,研究了一类带有P-Lapalcian算子Caputo型非线性分数阶微分方程的正解问题;并通过构造第二变元的Lipschitz条件,在P-Lapalcian算子参数取不同值的范围下,分别证明了边值问题正解的存在性与唯一性.     

一类具p-laplacian算子的含积分边界条件的分数阶微分方程边值问题解的存在性

利用Green函数的性质、u_0-边界函数、不动点指数定理及锥压缩与锥拉升不动点定理,研究一类具p-laplacian算子的含积分边界条件的微分方程边值问题解的存在性.     

Differentiation of Singular Integrals with Piecewise-Continuous Density and Boundary Values of Derivatives of a Cauchy-Type Integral

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the nth-order derivatives of a singular Cauchy integral and for the boundary values of the nth-order derivatives of a Cauchy-type integral are obtained.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 222–229, February, 2005.