Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

Uniqueness of Oblique Derivative Problems for Second Order Equations of Mixed Type with Nonsmooth Degenerate Line

In [], the authors discussed the Tricomi problem of second order mixed equations with nonsmooth degenerate line, but they only consider some special mixed equations. In [], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equations with nonsmooth degenerate line. The present article deals with oblique derivative problems for general second order mixed equations with nonsmooth parabolic degenerate line, and prove the uniqueness of solutions of the problems. We first give the formulation of the problems for the equations, and then prove the uniqueness of solutions for the above problems.     

Persistent regional null controllability of some degenerate wave equations

This paper is addressed to a study of the persistent regional null controllability problems for one‐dimensional linear degenerate wave equations through a distributed controller. Different from non‐degenerate wave equations, the classical null controllability results do not hold for some degenerate wave equations. Thus, persistent regional null controllability is introduced, which means finding a control such that the corresponding state of the degenerate wave equation may vanish in a suitable subset of the space domain in a period of time. In order to solve this problem, we need to establish the regional null controllability for degenerate wave equations. This problem is reduced to a suitable observability problem of a linear degenerate wave equation. The key point is to choose a suitable multiplier in order to establish this observability inequality. Copyright © 2017 John Wiley & Sons, Ltd.     

Elliptic equations strongly degenerate at a point

In this paper we will mainly propose some problems for a class of degenerate elliptic equations, either linear or nonlinear. We will study some special cases of these problems and reveal some phenomena which may not have been noticed previously.     

Congested traffic dynamics, weak flows and very degenerate elliptic equations

Starting from a model of traffic congestion, we introduce a minimal-flow-like variational problem whose solution is characterized by a very degenerate elliptic PDE. We precisely investigate the relations between these two problems, which can be done by considering some weak notion of flow for a related ODE. We also prove regularity results for the degenerate elliptic PDE, which enables us in some cases to apply the DiPerna–Lions theory.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

Steady states of radially symmetric diffusion processes

This note is concerned with the steady states of some diffusion processes arising from some physical problems. We are interested in the discussion of radially symmetric steady states, which lead to a degenerate and singular equation. Under some natural structural condition, we obtain the existence and uniqueness of nonnegative solutions.     

线性规划多重最优解判别准则刍议

本构造了一些线性规划问题来探讨多重最优解的判别准则，补充了现行献中关于多重最优解判别准则描述的不足，并指出多重最优解判别准则在出现退化解时可能失效的例外情况。     

Direct and inverse problems for degenerate differential equations

We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on \((-\infty ,\infty )\). Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too.     

Boundary value problems for first-order mixed complex equations with degenerate curve

This article deals with some boundary value problems for quasilinear mixed (elliptic-hyperbolic) complex equations of first order with degenerate hyperbolic curve in a simply connected domain. First, we give the representation theorem and prove the uniqueness of solutions for the above boundary value problems. Then using the method of successive iteration, the existence of solutions for the above problems is proved.     

Sui problemi al contorno per i sistemi ellittici degeneri di ordine superiore

Boundary value problems for degenerate elliptic systems of higher order are studied and some existence-uniqueness theorems are shown.     

Residual Selection in A Projection Method for Convex Minimization Problems

We study a certain class of min-max optimization problems and we show that the solutions of such problems are, in some sense, degenerate, Two examples are given:minimization of the spectral radius of a symmetric matrix, minimization of the largest eigenvalue of a symmetric matrix     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

Steklov operators and semigroups in weighted spaces of continuous real functions

We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula.     

Liouville-Type Theorem for a Nonlinear Degenerate Parabolic System of Inequalities

In this paper, we establish some new Liouville-type results for solutions of nonlinear degenerate parabolic system of inequalities. Nonexistence of nontrivial global solutions to initial-value problems is studied by using scaling transformations and test functions.     

Existence and multiplicity results for classes of elliptic resonant problems

In this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems. We also establish several multiplicity results. The methods used here are based on combining the minimax methods and the Morse theory especially some new observations on the critical groups of a local linking-type degenerate critical point.     

Classe de compacite d’operateur intervenant dans une classe de problemes elliptiques degeneres

We compute here the class of compacity of those operators onL ² (Ω) the image of which belongs to some families of weighted Sobolev spaces. Such spaces are relevant for the study of some elliptic problems which degenerate at the boundary of Ω.     

Existence results for some Kirchhoff-Carrier problems

Analyzing the viscoelastic problem for small vibrations of elastic strings, Kirchhoff and Carrier proposed two different models of nonlinear partial differential equations. By combining these two models, we deal here with some nonlocal hyperbolic problems that cover a large class of Kirchhoff and Carrier type problems. The existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established. The proofs are based on the combination of the Schauder fixed point theorem with some asymptotic method.     

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.     

Cauchy Problems for Degenerate Hyperbolic Monge-Ampere Equations and Some Applications

In this paper, the Cauchy problems for a class of nonlinear degenerate hyperbolic Monge-Ampere equations are studied and some results on local isometric embedding in R^3 of two dimensional Riemannian manifolds with non positive curvature, are contained.