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.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

Boundary integral equations for screen problems in IR3

Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR³ via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.     

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C ^{1, α} on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C ^{2, α} on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.     

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in \mathbbR^N {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure.     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ?³. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.     

Boundary integral equations for mixed Dirichlet,Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.     

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ₂ and given Neumann data on the remaining part Γ₂ of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ₂ and the also desired values of the normal derivatie on Γ₁. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ₁ ∩ Γ₂ even for C^∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ₂ providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ₁ and Γ₂ augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).     

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Abstract Elliptic Equations with Integral Boundary Conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in $L_{p}$ spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed $L_{\mathbf{p}}$ norm.     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

On closed boundary value problems for equations of mixed elliptic‐hyperbolic type

For partial differential equations of mixed elliptic‐hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet‐conormal types. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a ? b ? c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. © 2006 Wiley Periodicals, Inc.     

The Spin-Coating Process: Analysis of the Free Boundary Value Problem

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.     

Boundary difference-integral equation method and its error estimates for second order hyperbolic partial differential equation

Combining difference method and boundary integral equation method, we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain inR ³ and obtain the error estimates of the approximate solution in energy norm and local maximum norm.China State Major Key Project for Basic Researches.     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.     

Principal Eigenvalues for Isaacs Operators with Neumann Boundary Conditions

In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded C ² domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.