Rellich Identity Research Articles

Rellich identities for the Hilbert transform

We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new L2-weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted L2 in terms of a weight constant associated to the Helson-Szegö theorem.

Stability of grating diffraction problems for plane wave incidence: Explicit dependence on wavenumbers and incident angles

Suppose a time-harmonic plane wave is incident onto an impenetrable grating profile of Dirichlet or Impedance kind in two dimensions. The grating profile is assumed to be given by a Lipschitz function. We derive a stability estimate of this grating diffraction problem using the variational method with a transparent boundary condition. An explicit dependence of solutions on the incident wavenumber and the incident angle is obtained, which carries over to transmission problems for penetrable gratings. Our approach relies heavily on using Rellich's identities in periodic structures.

Radial symmetry and partially overdetermined problems in a convex cone

AbstractWe obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin‐type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.

Hybridization of the rigorous coupled-wave approach with transformation optics for electromagnetic scattering by a surface-relief grating

We hybridized the rigorous coupled-wave approach (RCWA) with transformation optics to develop a hybrid coordinate-transform method for solving the time-harmonic Maxwell equations in a 2D domain containing a surface-relief grating. In order to prove that this method converges for the p-polarization state, we studied several different but related scattering problems. The imposition of generalized non-trapping conditions allowed us to prove a-priori estimates for these problems. To do this, we proved a Rellich identity and used density arguments to extend the estimates to more general problems. These a-priori estimates were then used to analyze the hybrid method. We obtained convergence rates with respect to two different parameters, the first being a slice thickness indicative of spatial discretization in the depth dimension, the second being the number of terms retained in the Rayleigh–Bloch expansions of the electric and magnetic field phasors with respect to the other dimension. Testing with a numerical example revealed faster convergence than our analysis predicted. The hybrid method does not suffer from the Gibbs phenomenon seen with the standard RCWA.

Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds

Abstract The simulation of the elastodynamics equations at high frequency suffers from the well-known pollution effect. We present a Petrov–Galerkin multiscale sub-grid correction method that remains pollution-free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the log ⁡ ( k ) \log(k) , 𝑘 being the wave number. Key to this method are polynomial-in-𝑘 bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-𝑘 bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in R 3 \mathbb{R}^{3} . Previous methods relied on variational techniques, Rellich identities, and geometric constraints. In the context of elastodynamics, these suffer from the need to hypothesize a Korn’s inequality on the boundary. The methods in this work are based on boundary integral operators and estimation of Green’s function’s derivatives dependence on 𝑘 and do not require this extra hypothesis. We also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.

Analysis of the Rigorous Coupled Wave Approach for [formula omitted]-polarized light in gratings

We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for s-polarized monochromatic incident light. The RCWA is widely used to solve electromagnetic boundary-value problems where the relative permittivity varies periodically in one direction, i.e., scattering by a grating. This semi-analytical approach expands all the electromagnetic field phasors as well as the relative permittivity as Fourier series in the spatial variable along the direction of periodicity, and also replaces the relative permittivity with a stairstep approximation along the direction normal to the direction of periodicity. Thus, there is error due to Fourier truncation and also due to the approximation of grating permittivity. We prove that the RCWA is a Galerkin scheme, which allows us to employ techniques borrowed from the Finite Element Method to analyze the error. An essential tool is a Rellich identity that shows that certain continuous problems have unique solutions that depend continuously on the data with a continuity constant having explicit dependence on the relative permittivity. We prove that the RCWA converges with an increasing number of retained Fourier modes and with a finer approximation of the grating interfaces. Numerical results show that our convergence results for increasing the number of retained Fourier modes are seen in practice, while our estimates of convergence in slice thickness are pessimistic.

A note on Kuttler\u2013Sigillito\u2019s inequalities

We provide several inequalities between eigenvalues of some classical eigenvalue problems on compact Riemannian manifolds with C^2 boundary. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of mathbb {R}^2 to the manifold setting.

The Neumann problem for higher order elliptic equations with symmetric coefficients

In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$ or the Sobolev space $$\dot{W}^2_{-1}$$ , in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.

Rellich–Christianson type identities for the Neumann data mass of Dirichlet eigenfunctions on polytopes

We consider the Dirichlet eigenvalue problem on a polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particularly simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson (Equidistribution of Neumann data mass on simplices and a simple inverse problem, ArXiv e-prints, 2017, Equidistribution of Neumann data mass on triangles. ArXiv e-prints, 2017).

On uniqueness in electromagnetic scattering from biperiodic structures

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all positive wave numbers.

SCATTERING OF PLANE ELASTIC WAVES BY THREE-DIMENSIONAL DIFFRACTION GRATINGS

The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. A radiation condition based on the Rayleigh expansion of the quasi-periodic solutions is presented. Existence of solutions in Sobolev spaces is established if the grating profile is a two-dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function.

Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.

Electromagnetic Wave Scattering from Rough Penetrable Layers

We consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes, for instance, propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable nontrapping and smoothness conditions, this integral identity allows us to prove an a priori estimate for such a solution. A priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem.

ℎ𝑝-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we develop and analyze some interior penalty h p hp -discontinuous Galerkin ( h p hp -DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The proposed h p hp -DG methods are defined using a sesquilinear form which is not only mesh-dependent (or h h -dependent) but also degree-dependent (or p p -dependent). In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p p . Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed h p hp -discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number k k , sub-optimal order (with respect to h h and p p ) error estimates in the broken H 1 H^1 -norm and the L 2 L^2 -norm are derived without any mesh constraint. The error estimates as well as the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p − 2 ≤ C 0 k^3h^2p^{-2}\le C_0 by utilizing these stability and error estimates and using a stability-error iterative procedure, where C 0 C_0 is some constant independent of k k , h h , p p , and the penalty parameters. To overcome the difficulty caused by strong indefiniteness (and non-Hermitian nature) of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h h , the polynomial degree p p , the wave number k k , as well as all the penalty parameters for the numerical solutions.

Variational approach to scattering of plane elastic waves by diffraction gratings

The scattering of a time-harmonic plane elastic wave by a two-dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasiperiodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies. Copyright © 2010 John Wiley & Sons, Ltd.

A variational method for wave scattering from penetrable rough layers

We consider time-harmonic wave scattering from unbounded penetrable rough layers and provide existence theory for this problem via variational formulations. A rough layer is a model for a stratified and unbounded inhomogeneous medium where the refractive index varies. For our variational approach, the refractive index can be real or (partially) complex valued and is allowed to jump across interfaces. However, the index needs to satisfy a non-trapping condition, which requires, roughly speaking, monotonicity in the direction normal to the layer. In the half-space above and below the rough layer, we set up a radiation condition using the angular spectrum representation. Due to the unbounded setting, we establish integral formulas similar to Rellich's identity to obtain an a priori bound for a variational solution of the rough-layer scattering problem. This a priori bound is the basis for our existence result. We further provide regularity theory for the rough-layer scattering problem and give bounds on its frequency dependence. Our existence results hold for all wave numbers and apply to acoustic scattering in dimension two and three as well as to electromagnetic scattering for the electric and the magnetic mode.

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops and analyzes some interior penalty discontinuous Galerkin (IPDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and suboptimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed IPDG methods include the following: First, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To this end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [P. Cummings, Analysis of Finite Element Based Numerical Methods for Acoustic Waves, Elastic Waves and Fluid-Solid Interactions in the Frequency Domain, Ph.D. thesis, The University of Tennessee, Knoxville, TN, 2001], [P. Cummings and X. Feng, Math. Models Methods Appl. Sci., 16 (2006), pp. 139-160], [U. Hetmaniuk, Commun. Math. Sci., 5 (2007), pp. 665-678].

SHARP REGULARITY COEFFICIENT ESTIMATES FOR COMPLEX-VALUED ACOUSTIC AND ELASTIC HELMHOLTZ EQUATIONS

We consider complex-valued acoustic and elastic Helmholtz equations with the first order absorbing boundary condition in a star-shaped domain in ℜN for N ≥ 2. It is known that the elliptic regularity coefficients depend on the frequency ω, and have singularities for both zero and infinite frequency. In this paper, we obtain sharp estimates for the coefficients with respect to large frequencies. It is proved that the elliptic regularity coefficients are bounded by first or second order polynomials in ω for large ω. The crux of our analysis is to establish and make use of Rellich identities for the solutions to the acoustic and elastic Helmholtz equations. Our results improve the earlier estimates of Refs. 10 and 11, which were carried out based on layer potential representations of the solutions of the Helmholtz equations.

Some norm estimates for the solutions of a nonlinear diffusion problem

By means of Rellich's identity bounds for the spectrum of the nonlinear problem Δv+λe v=0 are derived and certain norms for the solutions are estimated.