Degree Theory for Second Order Nonlinear Elliptic Operators and its Applications

We introduce, along the lines of [4], an integer valued degree for second order fully nonlinear elliptic operators which is invariant under homotopy within elliptic operators. We also give some applications to the bifurcation problem for nonlinear elliptic equations. Applications to the existence of solutions of certain fully nonlinear elliptic equations on compact manifolds can be found in [7].     

First Order LSFEM for Second Order Hyperbolic Problems

In the last years the interest in least squares finite element methods has grown due to some interesting properties of these methods (cf. the monographs [1], [2]). While for many elliptic problems the theoretical background has been established, only a few articles analyse least squares methods for transient problems. In this article some new stability estimates for the least squares method for the wave equation in 1 and 2 dimensions are derived. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parabolic First and Second Order Differential Equations

We are concerned with an inverse problem for a first-order linear evolution equation. Moreover, a complete second-order evolution equation will be considered, too. We indicate sufficient conditions for existence and uniqueness of a solution. All the results apply well to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

Superparabolic Functions Related to Second Order Hypoelliptic Operators

In this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions.     

Weak Solutions of Second Order Parabolic Equations in Noncylindrical Domains

Penalization method is used to get the existence of the weak solutions or parabolic equations in noncylindrical domains. The asympcotic decay and the existence and uniqueness of the periodic solutions are obtained as well.     

二阶微分算子的一个注记

此文利用黎曼几何的知识将二阶椭圆算子表示成为一个Schr(?)dinger算子．     

Block Hopscotch Procedures for Second Order Parabolic Differential Equations

The authors consider a general block implicit hopscotch procedurefor solving parabolic partial differencial equations. Stabilityand convergence analyses are given and an examination of thecomparative accuracy of various particular cases is given. Theusefulness of these algorithms in solving problems using (r)geometry is indicated and their extensions to general exteriorproblems using a peripheral approach is included.     

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

In this paper we study the homogenization of degenerate quasilinear parabolic equations: where a(t, y, a, λ) is periodic in (t, y).     

Meromorphic Continuation of Scattering Matrices: Long Range,Stark Case

Long range quantum mechanical scattering in the presence of a constant electric field of strength F > 0 is discussed. It is shown that the scattering matrix, as a function of energy, has a meromorphic continuation to the entire complex plane as a bounded operator on L(? ^n?1) where n is the space dimension. There is a marked contrast between this result and the comparable result in the Schrödinger case (F = 0). The scattering matrix is constructed using two Hilbert space wave operators and time dependent modified wave operators both and the constructions are compared.     

Implicit Difference Methods for Degenerate Hyperbolic Equations of Second Order

This paper is a sequel to [2]. A two parameter family of explicit and implicit schemes is constructed for the numerical solution of the degenerate hyperbolic equations of second order. We prove the existence and the uniqueness of the solutions of these schemes. Furthermore, we prove that these schemes are stable for the initial values and that the numerical solution is convergent to the unique generalized solution of the partial differential equation.     

二阶退化双曲型方程的Darboux型问题

本文讨论二阶退化双曲型方程的一些边值问题.文中先给出第一Darboux问题和一般斜微商边值问题的提法和解的表示式,然后使用复分析方法证明了上述问题解的存在唯一性.     

Long Range Scattering and Modified Wave Operators for the Maxwell–Schrödinger System II. The General Case

We study the theory of scattering for the Maxwell–Schr?dinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr?dinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr?dinger equation, which then becomes both nonlinear and nonlocal in time. The Schr?dinger function is then parametrized in terms of an amplitude and a phase satisfying a suitable auxiliary system, and the Cauchy problem for that system, with prescribed asymptotic behaviour determined by the asymptotic data, is solved by an energy method, thereby leading to solutions of the original system with prescribed asymptotic behaviour in time. This paper is the generalization of a previous paper with the same title. However it is entirely self contained and can be read without any previous knowledge of the latter. Submitted: November 7, 2006. Accepted: November 14, 2006.     

Viscosity Solutions to Second Order Parabolic PDEs on Riemannian Manifolds

In this work we consider viscosity solutions to second order parabolic PDEs u _t +F(t,x,u,du,d ² u)=0 defined on complete Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold M has nonnegative sectional curvature, we get the finest results. If one additionally requires F to depend on d ² u in a uniformly continuous manner, the assumptions on curvature can be thrown away.     

Homogenization Estimates of Nondivergence Elliptic Operators of Second Order

Mathematical Notes - Questions related to the homogenization of the Dirichlet problem for nondivergence elliptic equations of second order with rapidly oscillating $$varepsilon$$ -periodic...     

Sampling Integrodifferential Transforms Arising from Second Order Differential Operators

We give sampling theorems associated with boundary value problems whose differential equations are of the form M(y) = λS(y), where M and S are differential expressions of the second and first order respectively and the eigenvalue parameter may appear in the boundary conditions. The class of the sampled functions is not a class of integral transforms as is the case in the classical sampling theory, but it is a class of integrodifferential transforms. We use solutions of the problem as well as Green's function to derive two sampling theorems.