Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

一类带奇异低阶项椭圆型方程的正则性

本文得到了一类带奇异低阶项椭圆方程的一个正则性结果.方程形式为Lu+ vu=μ,其中L为二阶散度型椭圆算子,v属于某一Morrey空间,μ为一非负Radon 测度.     

Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

Here we give an existence and uniqueness result of a renormalized solution for a class of nonlinear parabolic equations \(\displaystyle {\partial b(u) \over \partial t} - \mathrm{div}(a(x,t,\nabla u))+\mathrm{div}(\Phi (x,t, u))=\mu \), where the right side is a measure data, b is a strictly increasing \(C^1\)-function, \(- \mathrm{div}(a(x,t,\nabla u))\) is a Leray–Lions type operator with growth \(|\nabla u|^{p-1}\) in \(\nabla u\) and \(\Phi (x,t, u)\) is a nonlinear lower order term.     

Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem by means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.     

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   

间断系数的二阶椭圆型方程解的正则性

该文研究的问题源自于生物学与物理学中具有间断介电系数的静电场。作者以拟微分算子为主要工具讨论具间断系数的半线性二阶椭圆型方程解的存在性和正则性 。     

Nonlinear Anisotropic Elliptic and Parabolic Equations in R<Superscript>N</Superscript> with Advection and Lower Order Terms and Locally Integrable Data

We prove existence and regularity results for distributional solutions in R^N for nonlinear elliptic and parabolic equations with general anisotropic diffusivities as well as advection and lower-order terms that satisfy appropriate growth conditions. The data are assumed to be merely locally integrable. Mathematics Subject Classifications (2000) 35J60, 35K55.This work was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. This work was done while M. Bendahmane visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

带梯度项的椭圆型方程边界爆破解的高阶估计

In this paper,the higher order asymptotic behaviors of boundary blow-up solutions to the equation■in bounded smooth domain■are systematically investigated for p and q.The second and third order boundary behaviours of the equation are derived.The results show the role of the mean curvature of the boundary■and its gradient in the high order asymptotic expansions of the solutions.     

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．     

双参数奇摄动高阶半线性椭圆型方程的渐近解

讨论了一类具有双参数的高阶半线性奇摄动椭圆型方程的边值问题.在适当的假设下,构造了解的形式渐近展开式,并且利用不动点定理,研究了边值问题解的存在性和渐近性态.     

Symmetry Results for Positive Solutions of Some Elliptic Equations on Manifolds

We adapt the method of moving planes in order to obtainsymmetry results for positive solutions of some elliptic equations onmanifolds, in the case where our problem satisfies certain symmetryhypothesis. We also obtain monotonicity results using the slidingmethod.     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

二阶非线性椭圆型微分方程解的Sturm比较定理

本文通过对一类含阻尼项的椭圆型微分方程建立几个微分不等式,得到几个新的Sturm型比较定理,将经典的Sturm比较定理推广到含阻尼项椭圆型微分方程,并给出一些具体应用的例子.     

一类拟线性椭圆型方程下解的先验估计

本文在一定的结构条件下,对如下形状的拟线性椭圆型方程∫Ｇｖ·Ａ（ｘ,ｕ,ｕ）＋ｖＢ（ｘ,ｕ,ｕ）｛｝ｄｘ＝０,ｖ∈Ｗ１ｐ（Ｇ）的下解作出了新的先验估计,发展了Ｇｉｌｂａｒｇ＿Ｔｒｕｄｉｎｇｅｒ的某些结果·     

Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-\Delta u + b(x)|\nabla u| = c(x)u}$ in exterior domains of ${\mathbb{R}^N}$ . We show that if lim ${{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}$ and ${-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.     

Lp-estimates for the Strong Solutions of Elliptic Equations of Nondivergent Type

We investigate the second derivatives L^p-estimates for the strong solutions of second order linear elliptic equations in nondivergencc form Lu = f in the case in which the leading coefficients of L are not continuous. The L^p-estimates for small p are obtained if L is uniformly elliptic. Furthermore, if the leading coefficients of L belong to W^{1,n}, then we get the second derivatives L^p-estimates for large p. The existence of the strong solutions of the homogeneous Dirichlet problem is also considered.     

泛函极小与非线性椭圆方程解的局部正则性

利用Giaquinta和Giusti的嵌入不等式和Sobolev空间方法,证明了在更一般条件下的变分问题中的泛函极小与非线性椭圆方程弱解的局部正则性.     

二阶椭圆偏微分方程的Pinching-估计

Pinching-估计是研究解的凸性的一种重要方法,主要给出了半线性二阶椭圆偏微分方程的Pinching-估计,并将其推广到一类完全非线性二阶椭圆偏微分方程.