四阶椭圆方程的弱解(英文)

本文研究了一个四阶椭圆方程解的存在性问题.利用山路定理和喷泉定理,结合变分方法,获得了该问题弱解的几个存在性定理,推广了现有的一些结果.     

INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL

This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor model. We first obtain the global existence of weak solutions for the full model equation by employing the Galerkin's approximation method.Secondly, for a slightly simplified model, we show the existence and uniqueness of global strong solutions via the Banach's fixed point theorem and vanishing viscosity method.     

一类具有p-Laplacian的非线性特征值问题

利用变分法中的一个三临界点定理,在适当的假设下,研究了一类具有p-Laplacian的特征值Dirichlet问题至少存在三个弱解的充分条件.     

具有Dirichlet边界条件的一类拟线性椭圆方程组的多重解的存在性

运用Ricceri三临界点定理,研究了一类具有Dirichlet边界条件的拟线性椭圆方程组问题,证明了该方程组在其非线性项满足某些新的条件时至少存在三个解.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients

We prove an existence theorem for weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. A weak solution of an equation is understood as a weak solution of a stochastic differential inclusion constructed on the basis of the equation. We derive conditions providing the absence of blow-up in weak solutions.     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

Finite-difference scheme for two-scale homogenized equations of one-dimensional motion of a thermoviscoelastic Voigt-type body

The Bakhvalov-Eglit two-scale homogenized equations are used to describe the motion of layered periodic compressible media with rapidly oscillating data. A new finite-difference scheme for a system of such equations is proposed and analyzed in the case of a thermoviscoelastic Voigt-type body. A priori estimates of solutions are derived for nonsmooth data. The existence and uniqueness of discrete solutions are established. A theorem is proved on the convergence of a subsequence of discrete solutions to a weak solution of the problem under study. Simultaneously, a new theorem on the existence of global weak solutions is deduced.     

Three solutions for a class of quasilinear elliptic equation involving the p − q‐Laplace operator

The existence of at least three weak solutions is established for a class of quasilinear elliptic equation involving the p ? q‐Laplace operator with Dirichlet boundary condition. The technical approach is mainly on the basis of a three critical points theorem due to Ricceri. Copyright © 2013 John Wiley & Sons, Ltd.     

The existence of solutions for a class of semilinear elliptic systems

In this paper, some new existence theorems of weak solutions for a class of semilinear elliptic systems are obtained by means of the local linking theorem and the saddle point theorem.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

On the Existence of Solutions of Quasivariational Inclusion Problems

Quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.     

Weak solutions for contact models involving a class of generalized materials

We consider a contact model for a class of generalized materials whose behaviour is described by means of a bipotential. Using the theory of the bipotentials we deliver a weak formulation consisting of a system of three variational inequalities. The unknown is a triple having as components the displacement field, the Cauchy stress tensor and a Lagrange multiplier related to the friction force on the frictional contact zone. We investigate the existence of the weak solutions by using a fixed point theorem for set-valued mappings and a minimization technique.     

输液管非线性运动方程弱解的存在唯一性

本文利用Faedo-Galerkin法证明了一简支且两端不可移动的细长输液直管非线性运动方程弱解的存在唯一性。     

Global weak solutions for a modified two-component Camassa-Holm equation

We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa-Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space-time higher integrability estimates on the first-order derivatives of approximation solutions.     

Existence of periodic solutions for the thermistor problem with the Joule–Thomson effect

This paper deals with the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the Joule–Thomson effect. The periodicity of solutions is established utilizing the classical Lax–Milgram theorem and the Schauder fixed point theorem.      

Existence and multiplicity of solutions for Kirchhoff type equations

In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion,discontinuous coefficients,and a partly degenerate diffusion operator

We obtain estimates for functionals of solutions of stochastic differential equations with standard and fractional Brownian motion. We prove a theorem on the existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator.     

Elliptic problems involving the p-Laplacian in ℝN with locally Lipschitz functional

We study the existence of infinitely many weak solutions for the elliptic type equation via a general nonsmooth critical point theorem.     

On the statistical solutions of the two-dimensional,periodic Euler equation

We investigate the problem to define statistical solutions for the two-dimensional, periodic Euler Equation and prove a theorem of global existence and uniqueness for a regularized version. Moreover, we deduce the existence of the Euler flow in a weak form.