In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component i...In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component is a Kirchhoff type wave equation with nonlinear time dependent localized dissipation which is effective only on a neighborhood of certain part of the boundary, while the other is a Kirchhoff type wave equation with nonlinear memory.展开更多
In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitabl...In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.展开更多
In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component bei...In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component being a Kirchhoff type wave equation with time dependent localized dissipation which is effective only on a neighborhood of certain part of boundary, while the other being a Kirchhoff type viscoelastic wave equation with nonlinear memory展开更多
In this article we consider a transmission problem with memory in a bounded domain and varying delay term in the first equation. Under suitable assumptions on the weight of the damping and the weight of the delay, we ...In this article we consider a transmission problem with memory in a bounded domain and varying delay term in the first equation. Under suitable assumptions on the weight of the damping and the weight of the delay, we show the exponential sta- bility of the solution by introducing a suitable Lyapunov functional.展开更多
We give an alternative proof of a recent result in[1]by Caffarelli,Soria-Carro,and Stinga about the C^(1,α)regularity of weak solutions to transmission problems with C^(1,α)interfaces.Our proof does not use the mean...We give an alternative proof of a recent result in[1]by Caffarelli,Soria-Carro,and Stinga about the C^(1,α)regularity of weak solutions to transmission problems with C^(1,α)interfaces.Our proof does not use the mean value property or the maximum principle,and also works for more general elliptic systems with variable coefficients.This answers a question raised in[1].Some extensions to C^(1,Dini)interfaces and to domains with multiple sub-domains are also discussed.展开更多
This paper considers the inverse acoustic wave scattering by a bounded penetrable obstacle with a conductive boundary condition.We will show that the penetrable scatterer can be uniquely determined by its far-field pa...This paper considers the inverse acoustic wave scattering by a bounded penetrable obstacle with a conductive boundary condition.We will show that the penetrable scatterer can be uniquely determined by its far-field pattern of the scattered field for all incident plane waves at a fixed wave number.In the first part of this paper,adequate preparations for the main uniqueness result are made.We establish the mixed reciprocity relation between the far-field pattern corresponding to point sources and the scattered field corresponding to plane waves.Then the well-posedness of a modified interior transmission problem is deeply investigated by the variational method.Finally,the a priori estimates of solutions to the general transmission problem with boundary data in L^(p)(δΩ)(1<p<2)are proven by the boundary integral equation method.In the second part of this paper,we give a novel proof on the uniqueness of the inverse conductive scattering problem.展开更多
This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the sca...This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.展开更多
In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed t...In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small,certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.展开更多
We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust ...We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method （FEM） on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.展开更多
In this paper, we analyze the effect of both deter- ministic and random perturbations of a regular multi-layered elastic structure on its stop band properties. The tool of choice is the transfer matrix method, which i...In this paper, we analyze the effect of both deter- ministic and random perturbations of a regular multi-layered elastic structure on its stop band properties. The tool of choice is the transfer matrix method, which is both versatile and easy to implement. In both cases, we find that the stop-bands widen. We observe the appearance of very narrow pass-bands within the stop-bands, which can be observed in other instances in optics.展开更多
The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method.To use the abstract approximation theory for holomorphic ope...The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method.To use the abstract approximation theory for holomorphic operator functions,we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero.The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator.The spectral indicator method is employed to compute the transmission eigenvalues.Extensive numerical examples are presented to validate the theory.展开更多
The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission ...The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission eigenvalue problem is discretized by the Mor- ley element. For the resulting quadratic eigenvalue problem, a recursive integral method is used to compute real and complex eigenvalues in prescribed regions in the complex plane. Numerical examples are presented to demonstrate the effectiveness of the proposed method.展开更多
This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boun...This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boundary, the author establishes the uniform energy decay rate for the problem.展开更多
We develop a non-overlapping domain decomposition method(DDM)for scalar wave scattering by periodic layered media.Our approach relies on robust boun-dary-integral equation formulations of Robin-to-Robin(RtR)maps throu...We develop a non-overlapping domain decomposition method(DDM)for scalar wave scattering by periodic layered media.Our approach relies on robust boun-dary-integral equation formulations of Robin-to-Robin(RtR)maps throughout the frequency spectrum,including cutoff(or Wood)frequencies.We overcome the obsta-cle of non-convergent quasi-periodic Green functions at these frequencies by incor-porating newly introduced shifted Green functions.Using the latter in the defini-tion of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators.We develop Nystr̈om discretizations of the RtR maps that rely on trigonometric interpolation,singularity resolution,and fast convergent windowed quasi-periodic Green functions.We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully.We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.展开更多
文摘In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component is a Kirchhoff type wave equation with nonlinear time dependent localized dissipation which is effective only on a neighborhood of certain part of the boundary, while the other is a Kirchhoff type wave equation with nonlinear memory.
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(20110007870)
文摘In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.
文摘In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component being a Kirchhoff type wave equation with time dependent localized dissipation which is effective only on a neighborhood of certain part of boundary, while the other being a Kirchhoff type viscoelastic wave equation with nonlinear memory
文摘In this article we consider a transmission problem with memory in a bounded domain and varying delay term in the first equation. Under suitable assumptions on the weight of the damping and the weight of the delay, we show the exponential sta- bility of the solution by introducing a suitable Lyapunov functional.
基金supported by the Simons Foundation,grant No.709545。
文摘We give an alternative proof of a recent result in[1]by Caffarelli,Soria-Carro,and Stinga about the C^(1,α)regularity of weak solutions to transmission problems with C^(1,α)interfaces.Our proof does not use the mean value property or the maximum principle,and also works for more general elliptic systems with variable coefficients.This answers a question raised in[1].Some extensions to C^(1,Dini)interfaces and to domains with multiple sub-domains are also discussed.
文摘This paper considers the inverse acoustic wave scattering by a bounded penetrable obstacle with a conductive boundary condition.We will show that the penetrable scatterer can be uniquely determined by its far-field pattern of the scattered field for all incident plane waves at a fixed wave number.In the first part of this paper,adequate preparations for the main uniqueness result are made.We establish the mixed reciprocity relation between the far-field pattern corresponding to point sources and the scattered field corresponding to plane waves.Then the well-posedness of a modified interior transmission problem is deeply investigated by the variational method.Finally,the a priori estimates of solutions to the general transmission problem with boundary data in L^(p)(δΩ)(1<p<2)are proven by the boundary integral equation method.In the second part of this paper,we give a novel proof on the uniqueness of the inverse conductive scattering problem.
基金The Major State Basic Research Development Program Grant (2005CB321701)the Heilongjiang Education Committee Grant (11551364) of China
文摘This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.
基金supported by the Jilin Natural Science Foundation,China(No.20220101040JC)the National Natural Science Foundation of China(No.12271207)+2 种基金supported by the Hong Kong RGC General Research Funds(projects 11311122,12301420 and 11300821)the NSFC/RGC Joint Research Fund(project N-CityU 101/21)the France-Hong Kong ANR/RGC Joint Research Grant,A_CityU203/19.
文摘In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small,certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.
文摘We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method （FEM） on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.
基金a Marie Curie Transfer of Knowledge Fellowship of the European Community's Sixth Framework Programme under contract number(MTKD-CT-2004-509809)
文摘In this paper, we analyze the effect of both deter- ministic and random perturbations of a regular multi-layered elastic structure on its stop band properties. The tool of choice is the transfer matrix method, which is both versatile and easy to implement. In both cases, we find that the stop-bands widen. We observe the appearance of very narrow pass-bands within the stop-bands, which can be observed in other instances in optics.
基金supported in part by the National Natural Science Foundation of China with Grant No.11901295Natural Science Foundation of Jiangsu Province under BK20190431+1 种基金partially supported by the National Natural Science Foundation of China with Grant No.11971468Beijing Natural Science Foundation Z200003,Z210001.
文摘The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method.To use the abstract approximation theory for holomorphic operator functions,we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero.The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator.The spectral indicator method is employed to compute the transmission eigenvalues.Extensive numerical examples are presented to validate the theory.
文摘The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission eigenvalue problem is discretized by the Mor- ley element. For the resulting quadratic eigenvalue problem, a recursive integral method is used to compute real and complex eigenvalues in prescribed regions in the complex plane. Numerical examples are presented to demonstrate the effectiveness of the proposed method.
基金This research is supported by the National Natural Science Foundation of China under Grant Nos. 10571161 and 60774014.
文摘This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boundary, the author establishes the uniform energy decay rate for the problem.
文摘We develop a non-overlapping domain decomposition method(DDM)for scalar wave scattering by periodic layered media.Our approach relies on robust boun-dary-integral equation formulations of Robin-to-Robin(RtR)maps throughout the frequency spectrum,including cutoff(or Wood)frequencies.We overcome the obsta-cle of non-convergent quasi-periodic Green functions at these frequencies by incor-porating newly introduced shifted Green functions.Using the latter in the defini-tion of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators.We develop Nystr̈om discretizations of the RtR maps that rely on trigonometric interpolation,singularity resolution,and fast convergent windowed quasi-periodic Green functions.We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully.We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.
