Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension.The set of singular points consists of some singular lines and some isolated singular points.It is proved that near a singular line or a singular point,each weak solution can be decomposed into two parts,a singular part and a regular part.The singular parts are some finite sum of particular solutions to some simpler equations,and the regular parts are bounded in some norms,which are slightly weaker than that in the Sobolev space H~2.     

Two Dimensional Interface Problems for Elliptic Equations

We study the structure of solutions to the interface problems for second order quasi-linear elliptic partial differential equations in two dimensional space. We prove that each weak solution can be decomposed into two parts near singular points, a finite sum of functions in the form of cr^α log^m rφ(θ) and a regular one w. The coefficients c and the C^{1,α} norm of w depend on the H¹-norm and the C^{º,α}-norm of the solution, and the equation only.     

二维线性色散方程的色散量子化现象

研究了定义在平面有界矩形区域的二维线性KdV方程和二维线性Schr?dinger方程的色散量子化现象.证明了在有理时刻,方程周期初边值问题的解是初值条件的线性组合,而在无理时刻,解呈现类分形,连续不可微的状态.     

Asymptotics and Blow-up for Mass Critical Nonlinear Dispersive Equations

The author considers mass critical nonlinear Schr（o）dinger and Korteweg-de Vries equations.A review on results related to the blow-up of solution of these equations is given.     

Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Interface Problems for Elliptic Systems

Interface problems for elliptic systems of second order partial differential equations are studied. The main result is that the solution in the neighborhood of the singular point can be divided into two parts one of which is a solution to the homogeneous system with constant coefficients, and the other one possesses higher regularity.     

Inverse Problems for Parabolic Equations

The determination of variable filtration through a layer is an important and actual problem in filtration theory. This problem has been examined in some particular cases, for instance in the papers [1, 2].     

变系数的一般型发展方程的色散估计

本文研究的是带变系数的一般型线性发展方程.首先建立了其基本解的一系列色散估计:Kato光滑型估计,极大函数估计及Strichartz估计.最后应用这些估计研究了一些非自治非线性色散方程的初值问题在H~s(R)空间中的局部可解性.     

An Unconditionally Stable Difference Approximation for a Class of Nonlinear Dispersive Equations

An unconditionally stable leapfrog finite difference scheme for a class of nonlinear dispersive equations is presented and analyzed. The solvability of the difference equation which is a tridiagonal circular linear system is discussed. Moreover, the convergence and stability of the difference scheme are also investigated by a standard argument so that more difficult priori estimations are avoided. Finally, numerical examples are given.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

High Order Regularity for Interface Problems

Interface problems for second order elliptic partial differential equations are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a linear combination of some particular solutions,and the other one belongs to H where s≥2 is arbitrary.     

Boundary-Value Problems for Second-Order Equations and for Systems of First-Order Equations

The oblique derivative problem for harmonic functions under violation of the Shapiro–Lopatinsky condition is considered as well as some multi-dimensional analogues of the Cauchy–Riemann system. These problems are reduced to nonelliptic pseudo-differential equations. A method generalizing the regularization of singular integral equations is also presented.     

On Inverse Problems for Semiconductor Equations

This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is an inhomogeneity in the PDE model (doping profile). For a particular type of measurement (related to the voltage-current map) we consider special cases of drift-diffusion equations, where the inverse problems reduces to a classical inverse conductivity problem. A numerical experiment is presented for one of these special situations (linearized unipolar case).Lecture held by P.A. Markowich in the Seminario Matematico e Fisico on April 5, 2004Received: June, 2004     

Numerical Locking Problems for Parabolic Equations

We investigate the time evolution of a previous elliptic model. We study a parabolic model concerning the “numerical locking” phenomenon, namely a model for three-dimensional curved rods clamped at the endpoints. We show that the convergence of the numerical schemes is independent of the discretization parameters and the thickness of the rod. We present corresponding numerical experiments and we make a comparison with the results obtained for the hyperbolic model.     

Numerical Locking Problems for Hyperbolic Equations

We investigate a time-dependent model concerning the “locking phenomenon,” namely a model for three-dimensional clamped curved rods. We show that the convergence of the numerical schemes is independent of the discretization parameters and the thickness of the rod. We present corresponding numerical experiments.