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
By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guara...By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.展开更多
The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the o...The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.展开更多
In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate pa...In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.展开更多
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 approx...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.展开更多
This paper discuss the existence of bifurcation point of positive solutions for the fully nonlinear elliptic equations involving super-critical Soboley exponent which include semilinear, MongeAmpere and Hessian equati...This paper discuss the existence of bifurcation point of positive solutions for the fully nonlinear elliptic equations involving super-critical Soboley exponent which include semilinear, MongeAmpere and Hessian equations as its examples, by setting abstract bifurcation theorem via the topological degree theory.展开更多
In this paper,we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients.We begin with an approach given by N.V.Krylov to parabolic equations in the whole spac...In this paper,we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients.We begin with an approach given by N.V.Krylov to parabolic equations in the whole space with VMOx coefficients.We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions,weighted Lp estimates with Muckenhoupt(Ap)weights,non-local elliptic and parabolic equations,as well as fully nonlinear elliptic and parabolic equations.展开更多
In this paper,we establish global C2 estimates to the Neumann problem for a class of fully nonlinear elliptic equations.As an application,we prove the existence and uniqueness of k-admissible solutions to the Neumann ...In this paper,we establish global C2 estimates to the Neumann problem for a class of fully nonlinear elliptic equations.As an application,we prove the existence and uniqueness of k-admissible solutions to the Neumann problems.展开更多
We discuss the existence of global classical solution for the uniformly parabolic equation ■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b（x,t,u,u<sub>x</sub>）,（x,t）∈（-1,1）×...We discuss the existence of global classical solution for the uniformly parabolic equation ■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b（x,t,u,u<sub>x</sub>）,（x,t）∈（-1,1）×(0,T], u（±1,t）=0,u（x,0）=■（x）, where a is strongly nonlinear with respect to u<sub>xx</sub>and ■ is not necessarily small.We also deal with nonuniform case.展开更多
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.
文摘By the Schauder-Tychonoff fixed-point theorem, we inyestigate the existence and asymptotic behavior of positive radial solutions of fully nonlinear elliptic equations in Re. We give some sufficient conditions to guarantee the existence of bounded and unbounded radial solutions and consider the nonexistence of positive solution in Rn.
文摘The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.
文摘In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.
文摘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.
文摘This paper discuss the existence of bifurcation point of positive solutions for the fully nonlinear elliptic equations involving super-critical Soboley exponent which include semilinear, MongeAmpere and Hessian equations as its examples, by setting abstract bifurcation theorem via the topological degree theory.
文摘In this paper,we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients.We begin with an approach given by N.V.Krylov to parabolic equations in the whole space with VMOx coefficients.We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions,weighted Lp estimates with Muckenhoupt(Ap)weights,non-local elliptic and parabolic equations,as well as fully nonlinear elliptic and parabolic equations.
基金supported by NSFC Grant Nos.11721101 and 11871255.
文摘In this paper,we establish global C2 estimates to the Neumann problem for a class of fully nonlinear elliptic equations.As an application,we prove the existence and uniqueness of k-admissible solutions to the Neumann problems.
基金Supported by the Open Office of Mathematica Institute,Academia Sinica.
文摘We discuss the existence of global classical solution for the uniformly parabolic equation ■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b（x,t,u,u<sub>x</sub>）,（x,t）∈（-1,1）×(0,T], u（±1,t）=0,u（x,0）=■（x）, where a is strongly nonlinear with respect to u<sub>xx</sub>and ■ is not necessarily small.We also deal with nonuniform case.
