Comparison Principle for Viscosity Solutions with High Growth at Infinity

This paper is concerned with the comparison principle for viscosity solutions of the nonlinear elliptic equation F(Du, D²u} + |u|^{s-1}u =f in R^N, where f is uniformly continuous and F satisfies some conditions about p (p > 2}. We got the comparison principle for the viscosity solutions with some high growth at infinity, which relies on the relation between p and s.     

R^N中的非线性退化椭圆方程粘性解的存在性

本文研究了Ｒ＾Ｎ中的非线性退化椭圆型方程Ｆ（Ｄｕ，Ｄ＾２ｕ）＋ｕｓ＝ｆ的非负粘性解的存在性，其中ｓ〉０，Ｆ满足某些关于ｐ的条件，本文在下面的条件下证明了存在性；１．ｓ〉ｐ－１，ｆ在无穷远处不需要增长条件；２．０〈ｓ≤ｐ－１，ｆ在无穷远处具有某种增长条件。     

The Uniqueness of Viscosity Solutions of the Second Order Fully Nonlinear Elliptic Equations

Recently R. Jensen [1] has proved the uniqueness of viscosity solutions in W^{1,∞} of second order fully nonlinear elliptic equation F (D², Du, u) = 0. He does not assume F to be convex. In this paper we extend his result [1] to the case that F can be dependent on x, i. e. prove that the viscosity solutions in W^{1,∞} of the second order fully nonlinear elliptic equation F (D²u, Du, u, x) = 0 are unlique. We do not assume F to be convex either.     

Convergence of Rothe's method for fully nonlinear parabolic equations

Convergence of Rothe's method for the fully nonlinear parabolic equation u_t+F(D²u, Du, u, x, t)=0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Hölder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.     

一类一阶和二阶微分方程的渐近概周期解

对于一阶微分系统u′+F(u)=h(t),其中F为R~n上的严格单调算子,本文给出了其渐近概周期解存在和唯一的一个充分条件和一个必要条件.特别,对于一阶微分系统u′+▽Φ(u)=h(t),其中▽Φ代表R~N上凸函数Φ的梯度,讨论了其渐近概周期解存在和唯一的充分必要条件,并且把一些结果推广到了一类二阶方程.     

W2,ploc(\Omega)\cap C1,α(\bar Ω) Viscosity Solutions of Neumann Problems for Fully Nonlinear Elliptic Equations

In this paper we study fully nonlinear elliptic equations F(D²u, x) = 0 in Ω ⊂ R^n with Neumann boundary conditions \frac{∂u}{∂v} = a(x)u under the rather mild structure conditions and without the concavity condition. We establish the global C^{1,Ω} estimates and the interior W^{2,p} estimates for W^{2,q}(Ω) solutions (q > 2n) by introducing new independent variables, and moreover prove the existence of W^{2,p}_{loc}(Ω)∩ C^{1,α}(\bar \Omega} viscosity solutions by using the accretive operator methods, where p E (0, 2), α ∈ (0, 1}.     

C1,α Regularity of Viscosity Solutions of Fully Nonlinear Elliptic PDE Under Natural Structure Conditions

In this paper we are concemed with fully nonlinear elliptic equation F(x, u, Du, D²u) = 0. We establish the interior Lipschitz continuity and C^{1,α} regularity of viscosity solutions under natural structure conditions without differentiating the equation as usual, especially we give a new analytic Harnack inequality approach to C^{1,α} estimate for viscosity solutions instead of the geometric approach given by L. Caffarelli \& L. Wang and improve their results. Our structure conditions are rather mild.     

The Cauchy Problem for a Special System of Quasilinear Equations

We have obtained in this paper the existence of weak solutions to the Cauchy problem for a special system of quasillnear equations with physical interest of the form {\frac{∂}{∂t}(u + qz) + \frac{∂}{∂x}f(u) = 0 \frac{∂z}{∂t} + kφ(u)z = 0 for the assumed smooth function φ(u) by employing the viscosity method and the theory of compensated compactness.     

Solutions for the fractional Landau-Lifshitz equation

This article considers the dynamic equation of a reduced model for thin-film micromagnetics deduced by A. DeSimone, R.V. Kohn and F. Otto in [A. DeSimone, R.V. Kohn, F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1-53]. To derive the existence of weak solutions under periodical boundary condition, the authors first prove the existence of smooth solutions for the approximating equation, then prove the convergence of the viscosity solution when the viscosity term vanishes, which implies the existence of solutions for the original equation.     

A Note on C1,α Estimates for Solutions of Fully Nonlinear Elliptic Equations and Obstacle Problems

We deal with C^{1,α} interior estimates for solutions of fully nonlinear equation F(D²u, Du, x) = f(x) with the bounded gradient Du and a bounded f(x). Based on these estimates we obtain the existence of strong solutions of the obstacle problem for fully nonlinear elliptic equations under natural structure conditions.     

电报方程双周期解的极大值原理与强正性估计及应用

本文讨论非线性电报方程u_(tt)-u_(xx)+cu_t=F(t,x,u),(t,x)∈R~2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。     

二阶完全非线性蜕化抛物方程

本文研究了一个空间变量的二阶完全非线性蜕化抛物方程ｕｔ＝Ｆ（ｕｘｘ，ｕｘ，ｘ，ｔ）的第一边值问题。在仅要求Ｆ及其一阶导数满足结构条件的情形，给出了蜕化问题连续解的存在唯一性。这个工作将渗流方程的结果推广到非常一般的情形。     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

C1,α Regularity of Viscosity Solutions of Fully Nonlinear Parabolic PDE Under Natural Structure Conditions

In this paper, we concern the fully nonlinear parabolic equations u_t + F(x, t , u, Du, D² u) = 0. Under the natural structure conditions as that in [1], we obtain the C^{1,\alpha} estimates of the viscosity solutions.     

Periodic solutions for some Hamiltonian systems

In this paper, we study the existence of periodic solutions of some non-autonomous second order Hamiltonian systems $$\left\{\begin{array}{l}\ddot{u}(t)=\nabla{F\left(t,u(t)\right),\quad{}\mathrm{a.e.}\ t\in{[0,T]},}\\[2pt]u(0)-u(T)=\dot{u}{(0)}-\dot{u}{(T)}=0.\end{array}\right.$$ We obtain some new existence theorems by the least action principle.     

ON THE BOUNDED AND UNBOUNDED SOLUTIONS OF ONE DIMENSIONAL NONLINEAR REACTION－DIFFUSION PROBLEM

ＯＮＴＨＥＢＯＵＮＤＥＤＡＮＤＵＮＢＯＵＮＤＥＤＳＯＬＵＴＩＯＮＳＯＦＯＮＥＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＢＬＥＭ￥ＧＥＷＥＩＧＡＯＲ．Ｏ．ＷＥＢＥＲＡｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｂ...     

Existence of solutions to the Riemann problem for 2 × 2 conservation laws

We consider the Riemann problem for a class of 2?×?2 systems of conservation laws which do not satisfy the strictly hyperbolicity condition. Our main assumption is that the product of non-diagonal elements within the F?echet derivative (Jacobian) of the flux is nonnegative. By improving a vanishing viscosity approach, we establish the existence of solutions to the Riemann problem for those systems.     

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.     

Multi-valued solutions to fully nonlinear uniformly elliptic equations

In this paper, we discuss the existence and regularity of multi-valued viscosity solutions to fully nonlinear uniformly elliptic equations. We use the Perron method to prove the existence of bounded multi-valued viscosity solutions.     

p-Laplace方程Neumann边值问题的可解性

主要讨论了一维p-Laplace方程(φp(u′))′=f(t,u,u′),t∈(0,1))在Neumann边值条件u′(0)=0,u′(1)=0下边值问题解的存在性,其中φp(s)=|s|p-2s,s≠0.文中通过使用Leray-Schauder度原理,在适当的条件下,建立了对于p-Laplace方程在共振情形下Neumann边值问题解的存在性的充分条件.