Elliptic Equations with Degenerate Coercivity： Gradient Regularity

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is where for example, a(x,u)=(1+|u|)^−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.     

退化的非线性椭圆方程的径向解

利用打靶法讨论一类退化非线性椭圆型方程径向解的性态，得出了径向解的间断及不间断的结果．     

Regularity of the Solution to the Dirichlet Problem in Morrey Spaces

The aim of this paper is to study Morrey regularity of the solution to the Dirichlet problem for second order elliptic equation of the form Lu = -(a_{ij}u_x_i + b_j_u)_x_j + du = e + (f_j)x_j in a bounded open subset of Rn(n ≥ 3), where b_j , d, e, f_j are assumed to be in some Morrey spaces.     

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.     

Regularity of Solutions to Elliptic Equations with VMO Coefficients

The aim of this paper is to study the regularity of solutions to the Dirichlet problems for general second-order elliptic equations in Lebesgue and Morrey spaces. We consider both nondivergence and divergence forms and the coefficients of principle terms are assumed to be in VMO.     

Higher Integrability of the Gradient in Degenerate Elliptic Equations

We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta–Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution.     

具有低阶项的散度型椭圆方程的解在Morrey空间上的局部正则性

本文研究了具有低阶项的散度型椭圆方程-(a_(ij)u_(x_i))_(x_j)+b_iu_(x_i)-(d_ju)_(x_j)+cu= (f_j)_(x_j),a．e．x∈Ω的解在Morrey空间上的局部正则性,其中a_(ij)∈VMO∩L~∞(Ω),低阶项系数属于适当的Morrey空间．     

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.     

Regularity in Morrey Spaces of Strong Solutions to Nondivergence Elliptic Equations with VMO Coefficients

In this paper, by means of the theories of singular integrals and linear commutators, the authors establish the regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients.     

Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x.     

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.     

自然增长下非线性退化椭圆方程的优化Hlder连续指数

利用Moser-Nash迭代和稠密引理,得到了在自然增长下的非线性退化椭圆方程有界弱解具有某一Hlder指数的正则性;在已知数据的进一步正则性下,建立了具有任意γ满足0≤γ<κ的优化Hlder连续性指数,其中κ是A-调和函数的局部Hlder连续指数.     

Hölder Estimates for A Class of Degenerate Elliptic Equations

In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the Hölder type estimates for the weak solutions.     

一类含退化椭圆算子的Kirchhoff型方程解的存在性

利用临界点理论中的山路引理,证明一类含退化椭圆算子的Kirchhoff型方程在适当的假设条件下解的存在性,所得结论丰富和发展了已有文献的相关结果.     

一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献   

On Error Bounds for Approximation Schemes for Non-Convex Degenerate Elliptic Equations

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations.     

关于退化中立型微分方程的周期解

讨论了退化中立型微分方程的周期解问题,给出了周期解存在性的条件和二维退化中立型微分方程周期解存在的代数判据,并且举例说明了其应用.     

On the Dirichlet Problem for a System of Degenerate at a Point Elliptic Equations in the Class of Bounded Functions

We consider a weakly connected (by the lowest terms) system of elliptic equations of second order with the main part in the form of the Laplace operator, the order of which becomes degenerate at an interior point of the domain. We investigate a Dirichlet-type problem in the class of bounded Hölder vector functions. We obtain sufficient conditions for the existence and uniqueness of a solution.