Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

We obtain local C ^α, C ^1,α, and C ^2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.     

Fully Nonlinear Parabolic Equations and the Dini Condition

Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolic equations under the Dini condition, which improve and generalize a result due to Kovats, are obtained by the use of the approximation lemma. Received November 21, 2000, Accepted April 27, 2001     

Existence of Viscosity Solutions of Second Order Fully Nonlinear Elliptic Equations

We consider the problem of existence for viscosity solutions of seoond order fully nonlinear elliptic partial differential equations F(D²u, Du, u, z) = 0. We prove existence results for viscosity solutions in W^{1,∞} under assumptions that function F satisfies the natural structure conditions. We do not assume F is convex.     

State-Constrained Optimal Control Governed by Elliptic Differential Equations in Unbounded Domains

This work concerns the maximum principle for optimal control problems governed by elliptic differential equations in unbounded domains. Some state constraints are considered. This work was supported by National Natural Science Foundation of China, Grants 10401041 and 10471053.     

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

We consider the uniqueness of bounded continuous L^{3, ∞}-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here, L^{p, q} denotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in BC(?; L^{3, ∞}) within the class of solutions which have sufficiently small L^∞(L^{3, ∞})-norm. In this paper, we discuss another type of uniqueness theorem for solutions in BC(?; L^{3, ∞}) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations.     

完全非线性一致椭圆方程的外问题

利用Perron方法得到了完全非线性一致椭圆方程外问题具有渐近性质的粘性解的存在性.     

On the Existence of Solutions for Fully Nonlinear Elliptic Equations Under Relaxed Convexity Assumptions

We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D ² v, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D ² v| ≤K, where K is any given constant. For large |D ² v| some kind of relaxed convexity assumption with respect to D ² v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.     

一类非牛顿系统在三维无界区域中弱解的存在性(英文)

本文考虑一类非牛顿系统在三维无界区域中解的情况 ,证明了当f∈L2 ( 0 ,T ;H)时弱解的存在性 ;当f∈L∞( 0 ,∞ ;H)时弱解的全局存在性 .     

On the Rate of Convergence of Finite-Difference Approximations for Elliptic Isaacs Equations in Smooth Domains

We show that there exists an algebraic rate of convergence of solutions of finite-difference approximations for uniformly elliptic Isaacs equations in smooth bounded domains.     

椭圆型方程的边界正则性与Dini条件

笔者利用Caffarelli扰动方法,证明了Poisson方程和完全非线性一致椭圆型方程在Dini条件下其粘性解的边界正则性,从而给出了文[8]定理4.11不用位势理论的简化证明.     

A Counterexample to C 2,1 Regularity for Parabolic Fully Nonlinear Equations

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be C ^2,1.     

A Note on the Maximum Principle for Second-Order Elliptic Equations in General Domains

We make a further advance concerning the maximum principle for second-order elliptic operators. We investigate in particular a geometric condition, first considered by Berestycki Nirenberg Varadhan, that seems to be natural in view of the application of the boundary weak Harnack inequality, on which our argument is based. Setting it free from some technical assumptions, apparently needed in earlier papers, we significantly enlarge the class of unbounded domains where the maximum principle holds, compatibly with the first-order term.     

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C ^{1, α} on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C ^{2, α} on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.     

On Some Elliptic Problems in Unbounded Domains

The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution.     

Strong Solution of the Obstacle Problem for Fully Nonlinear Elliptic Equations

In this paper we obtain the existence of W^{2, ∞} solutions of the obstacle problems for fully nonlinear elliptic equations under more general structure conditions than those in [1] by using the mollifier approach, which is also extended in our discussion.     

Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully non-linear elliptic equations. Received: June 2006, Accepted: January 2007     

一类非线性散度形椭圆方程的最大值原理

文中运用 Hopf最大值原理 ,获得了具有 Dirichlet,Neumann和 Robin边界条件的非线性散度形椭圆方程 ( v( q) u,i) ,i+ w( q) f ( x,u) =0 ( q=| u| 2 ) 的解的函数的最大值原理 ,运用文中获得的最大值原理能够推出某些重要物理量的界的估计 .     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation u+f(,u)=0, in D with u=0 on D, where D is an unbounded domain in R ² with a compact nonempty boundary D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.     

REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ

ABSTRACT

In this second paper, we continue our study on the regularity of free boundaries for some fully nonlinear elliptic equations. Our result is if the free boundary is trapped in a sufficiently narrow strip formed by two Lipschitz graphs, then it is also a Lipschitz graph. Combining with the results in Part 1 (see Ref. [Wang]), the free boundary is C ^1,α.