This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up...This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up, , We get a prior estimate and existence of positive solutions of above equation.展开更多
We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these condition...We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.展开更多
Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）...Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）＋cu^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^n, the above equation is known as the thin film equation, which describes a steady state of the thin film （see Guo-Wei [Manuscripta Math., 120, 193-209 （2006）]）. The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233-256 （1991）], where the nonlinearity is the positive power of u.展开更多
The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0&...The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0<σ<1 is a real constant,and b(x,t) is a function which is C^2 in the x-variable and C^1 in the t-variable.展开更多
基金This research is supported by the National Natural Science Foundation of China.
文摘This paper is to study semilinear elliptic equation Lu = f(u) in Ω, Ω RN being a smooth bounded domain, with the third boundary value conditions =0, where operator, nonlinear function f(u) increase in the form of up, , We get a prior estimate and existence of positive solutions of above equation.
基金financed by the Alexander von Humboldt Foundationcontinued in March 2009 at the Mathematisches Forschungsinstitut Oberwolfach in the "Research in Pairs"program
文摘We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.
基金Partly supported by National Natural Science Foundation of China (Grant Nos. 1060106, 10811120558) the program for NCET
文摘Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）＋cu^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^n, the above equation is known as the thin film equation, which describes a steady state of the thin film （see Guo-Wei [Manuscripta Math., 120, 193-209 （2006）]）. The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233-256 （1991）], where the nonlinearity is the positive power of u.
基金supported by the Jiangsu Provincial Natural Science Foundation of China(No.BK20140804)the Fundamental Research Funds of the Central Universities(No.NS2014076)
文摘The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0<σ<1 is a real constant,and b(x,t) is a function which is C^2 in the x-variable and C^1 in the t-variable.
