In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-nes...In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.展开更多
The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with pi...The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.展开更多
The purpose of this paper is to prove existence of minimisers of the functional where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C_H^1(Ω\ K), α,β> 0,q≥1, g ∈ ...The purpose of this paper is to prove existence of minimisers of the functional where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C_H^1(Ω\ K), α,β> 0,q≥1, g ∈ Lq(Ω) ∩ L∞(Ω) and f : R2n→R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).展开更多
In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we ...In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 (A), 293-308 (1977)], [Walter, J., Math. Z., 133, 301-312 (1973)] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.展开更多
In this paper we consider the fully discrete local discontinuous Galerkin method, where the third order explicit Runge-Kutta time marching is coupled. For the one-dimensional time-dependent singularly perturbed proble...In this paper we consider the fully discrete local discontinuous Galerkin method, where the third order explicit Runge-Kutta time marching is coupled. For the one-dimensional time-dependent singularly perturbed problem with a boundary layer, we shall prove that the resulted scheme is not only of good behavior at the local stability, but also has the double-optimal local error estimate. It is to say, the convergence rate is optimal in both space and time, and the width of the cut-off subdomain is also nearly optimal, if the boundary condition at each intermediate stage is given in a proper way. Numerical experiments are also given.展开更多
In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its mo...In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.展开更多
Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends t...Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends to be a function on E (?)uch that it is harmonic in D and satisfies the Dirichletboundary condition at any regular boundary point of D. The relation between harmonic functions and theebaracteristic operator of the given process is discussed.展开更多
The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media.It is shown,in...The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media.It is shown,in particular,that,even if all interfaces are regular,the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources.However,when a simple and realistic constraint is enforced on these sources,requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial,among the problems here considered just those involving an interface between complementary materials remain ill-posed.These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present.These effects on finite element simulators are fully analyzed.Finally,we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical wellposed problems.展开更多
文摘In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.
文摘The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.
基金This work is supported by NNSF(10471063), Hunan NSF(03JJY4002) & Hunan Education Administration Item(03A011)
文摘The purpose of this paper is to prove existence of minimisers of the functional where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C_H^1(Ω\ K), α,β> 0,q≥1, g ∈ Lq(Ω) ∩ L∞(Ω) and f : R2n→R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).
文摘In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 (A), 293-308 (1977)], [Walter, J., Math. Z., 133, 301-312 (1973)] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.
文摘In this paper we consider the fully discrete local discontinuous Galerkin method, where the third order explicit Runge-Kutta time marching is coupled. For the one-dimensional time-dependent singularly perturbed problem with a boundary layer, we shall prove that the resulted scheme is not only of good behavior at the local stability, but also has the double-optimal local error estimate. It is to say, the convergence rate is optimal in both space and time, and the width of the cut-off subdomain is also nearly optimal, if the boundary condition at each intermediate stage is given in a proper way. Numerical experiments are also given.
文摘In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.
文摘Given a Markov process satisfying certain general type conditions,whose paths are notassumed to be continuous. Let D by an open subset of the state space E. Any bounded function defined on thecomplement of D extends to be a function on E (?)uch that it is harmonic in D and satisfies the Dirichletboundary condition at any regular boundary point of D. The relation between harmonic functions and theebaracteristic operator of the given process is discussed.
文摘The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media.It is shown,in particular,that,even if all interfaces are regular,the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources.However,when a simple and realistic constraint is enforced on these sources,requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial,among the problems here considered just those involving an interface between complementary materials remain ill-posed.These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present.These effects on finite element simulators are fully analyzed.Finally,we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical wellposed problems.