广义分布半群与退化发展方程的分布解

在Banach空间中引进了由有界线性算子引导的广义分布半群的新概念,并讨论了它的有关性质.在我们的方法中,广义分布半群的生成元可以不是稠定的.此外,还引进了退化发展方程在Laplace变换意义下的分布解,应用广义分布半群给出了退化发展方程分布解的构造性表达式.     

Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient _n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation-Part II

In this second part of the paper, through applying semigroup theory procedures, we study initial boundary problems associated with degenerate second-order differential operators of the form Lu(x) ? α(x) u″(x)+β(x)u′(x)+γ(x) u(x) in the framework of weighted continuous function spaces on an arbitrary real interval, when particular boundary conditions are imposed. By using the general results stated in the first part, we show that such operators, frequently occurring in Mathematical Finance, generate positive strongly continuous semigroups, which are, in turn, the transition semigroups associated with suitable Markov processes. Finally, an application to the Black-Scholes equation is discussed, as well.     

A Class of Degenerate Elliptic Equations

Hölder continuity of solutions is proved for a new class of degenerate divergent second-order elliptic equations with weight not satisfying the Muckenhoupt condition. There is no Harnack inequality and no Sobolev embedding theorems with higher summation exponent for these equations. As an example an equation is considered in a domain divided into two parts by a hyperplane. In each part the weight function is a power one, the powers are different, and their absolute values do not exceed the space dimension.     

非线性退化波方程的Riemann问题

研究了弹性力学中一退化波方程的Riemann问题．其应力函数非凸非凹,从而使得激波条件退化．通过引入广义激波条件下的退化激波,构造性地得到了各种情形下Riemann问题的整体解．     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Geometric Aspects of Degenerate Modulation Equations

Geometric aspects of degenerate modulation equations associated with spatially reversible systems are considered. Our primary observation is that stationary solutions of such equations always have a Poisson structure that is reminiscent of the equations governing the rigid body in mechanics. The Poisson structure is used to study the geometry of “spatial” phase space: A nontrivial Casimir of the Poisson structure provides a foliation of the phase space, spatially periodic states are given by critical points on level sets of the Casimir and stability type is given by the rate of change of the Casimir function. The bifurcation of spatially periodic states is then studied using singularity theory. The case where branches intersect transversely is treated in detail.     

Supersolutions to Degenerate Elliptic Equations on Quasi open Sets

A nonliner potenitial theory for the supersolutions of equations similar to the ^p-Laplacian is studied on quasi open setes. We also introduce Sobolev spaces W₁, p(U) and W_{1 p} ^o on quasi sets. The equivalence of several possible definitions of such spaces, as well as completencess, density results and an analogue of Baby's theorem, are proved. The basic probperties of supersolutions to the above equation on quasi open sets and corresponding finely A-superharmonic functions (a nonlinear version of Fugleds's concept) are investigated. We prove that each bounded finely superharmonic function is a fine supersoluan asymptotic corrdinate path along which an unbounded A-subharmonie function grows to indinity     

拟线性抛物方程的弱解

该文给出了拟线性退化抛物方程pa_t{u}+pa_x{f(u)}=pa_xx{A(u(x,t))}∈R^2_+×(0,+∞) ,u(x,0)=u_0(x),x∈R 一种弱解的新定义, 利用Div Curl引理证明了解的存在性．     

Degenerate Semilinear Elliptic Equations with Critical Exponent

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Irregular Solutions of Linear Degenerate Elliptic Equations

We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces.     

The Degenerate Venttsel Problem to Elliptic Equations

An existence theorem is proved for a quasilinear, degenerate, elliptic Venttsel BVP. Bibliography: 8 titles. To N. N. Uraltseva with gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 82–97.     

Degenerate Nonlinear Elliptic Equations Lacking in Compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|~(p-2)u) in R~N, where ▽_pu =|▽u|~(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.