Existence and Uniqueness of Solution for a Class of Nonlinear Degenerate Elliptic Equations

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations■,in the setting of the weighted Sobolev spaces.     

Nonlinear Degenerate Anisotropic Elliptic Equations with Variable Exponents and L1 Data

This paper is devoted to the study of a nonlinear anisotropic elliptic equation with degenerate coercivity, lower order term and L¹ datum in appropriate anisotropic variable exponents Sobolev spaces. We obtain the existence of distributional solutions.     

Morrey Regularity of Solutions to Degenerate Elliptic Equations in Rn

In this paper, we study the Morrey regularity of solutions to the de- generate elliptic equation -(a_{ij}u_{xi})_{xj} = -(f_j)_{xj} in R^n. For this purpose, we introduce four weighted Morrey spaces in R^n.     

Degenerate Nonlinear Boundary Value Problems for Fully Nonlinear Elliptic Equations of Second Order

In this paper, we will consider the degenerate nonlinear boundary value problems for nonlinear elliptic equations. We extend Lieberman & Trudinger's results from nondegenerate oblique derivative boundary values to degenerate boundary values.     

Dirichlet Problem for Degenerate Quasilinear Elliptic Equations Suggested by the Anisotropic Permeation

By Browder's pseudo-monotone operator theory and the techniques belonging to J. Leray and J. Lions, the existence theorem of the generalized solution of the Dirichlet problem for a strongly degenerate quasilincar elliptic equation has been proved in the anisotropic Sobolev space.     

Existence of Solutions for Some Quasilinear Degenerate Elliptic Inclusions in Weighted Sobolev Spaces

We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜(x, u, ?u)) + f(x)g(u) ∈ H(x, u, ?u), where 𝒜(x, u, ?u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H(x, u, ?u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained.     

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．     

Existence of Solutions of Some Degenerate Nonlinear Elliptic Fourth-order Equations with L 1 -data

This article deals with a class of degenerate nonlinear elliptic fourth-order equations with L 1 -right-hand sides. Equations of the given class have divergence form and their coefficients satisfy a strengthened ellipticity condition with two different weights associated, respectively, to the first- and the second-order derivatives of unknown function. Under suitable hypotheses on the weighted functions involved we establish solvability in a Sobolev space of Dirichlet problem for equations under consideration.     

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.     

Degenerate Semilinear Elliptic Equations with Critical Exponent

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The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampere equations is studied. Let Ω R^2 be smooth and convex. Suppose that u ∈ C^2（^-Ω） is a solution to the following problem： det（uij） = K（x）f（x,u, Du） in Ω with u = 0 on аΩ. Then u ∈ C^∞（f）） provided that f（x,u,p） is smooth and positive in ^-Ω × R × R^2, K〉0 in Ω and near αΩ, K = d^m ^-K, where d is the distance to αΩ, m some integer bigger than 1 and ^-K smooth and positive on ^-Ω.     

一类退化椭圆型方程边值问题的适定性

研究一类特殊退化椭圆型方程边值问题的适定性,该类问题与双曲空间中的极小图的Dirichlet问题,曲面的无穷小等距形变刚性问题等等的研究密切相关,而这类方程的特征形式在区域上是变号的,其适定性是值得深入讨论的.最后,得到这类边值问题的H~1弱解的存在性和唯一性.     

Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent

In this paper, we are concerned with a show the existence of a entropy solution to the obstacle problem associated with the equation of the type :$\begin{cases}Au+g(x,u,∇u) = f & {\rm in} & Ω \\ u=0 & {\rm on} & ∂Ω \end{cases}$where $\Omega$ is a bounded open subset of $\;\mathbb{R}^{N}$, $N\geq 2$, $A\,$ is an operator of Leray-Lions type acting from $\; W_{0}^{1,\overrightarrow{p}(.)} (\Omega,\ \overrightarrow{w}(.))\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'(.)} (\Omega,\ \overrightarrow{w}^*(.))$ and $\,L^1\,-\,$deta. The nonlinear term $\;g\,$: $\Omega\times \mathbb{R}\times \mathbb{R}^{N}\longrightarrow \mathbb{R} $ satisfying only some growth condition.     

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.     

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.     

An Identification Problem for First-Order Degenerate Differential Equations

We study a first-order identification problem in a Banach space. We discuss the nondegenerate and mainly the degenerate case. As a first step, suitable hypotheses on the involved closed linear operators are made in order to obtain unique solvability after reduction to a nondegenerate case; the general case is then handled with the help of new results on convolutions. Some applications to partial differential equations motivate this abstract approach.Communicated by I. GalliganiWork partially supported by MIUR (Ministero dell’ Istruzione, dell’ Università e dalla Ricerca), Project PRIN 2004011204 “Analisi Matematica nei Problemi Inversi,” and by the University of Bologna Funds for Selected Research Topics.     

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.