数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (1): 15-40.doi: 10.1016/S0252-9602(12)60003-4

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A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

 Bernd Kawohl1*, Nikolai Kutev2   

  1. 1.Mathematisches Institut, Universit¨at zu K¨oln, D 50923 K¨oln, Germany|2.Institute of Mathematics, Bulgarian Academy of Science, Acad. G. Bonchev Str. 8, 1113 Sofia, Bulgaria
  • 收稿日期:2011-09-02 出版日期:2012-01-20 发布日期:2012-01-20
  • 通讯作者: Bernd Kawohl,kawohl@math.uni-koeln.de E-mail:kawohl@math.uni-koeln.de; kutev@math.bas.bg

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

 Bernd Kawohl1*| Nikolai Kutev2   

  1. 1.Mathematisches Institut, Universit¨at zu K¨oln, D 50923 K¨oln, Germany|2.Institute of Mathematics, Bulgarian Academy of Science, Acad. G. Bonchev Str. 8, 1113 Sofia, Bulgaria
  • Received:2011-09-02 Online:2012-01-20 Published:2012-01-20
  • Contact: Bernd Kawohl,kawohl@math.uni-koeln.de E-mail:kawohl@math.uni-koeln.de; kutev@math.bas.bg

摘要:

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.

关键词: fully nonlinear elliptic equations, viscosity solutions, gradient estimates, gra-dient blow up

Abstract:

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.

Key words: fully nonlinear elliptic equations, viscosity solutions, gradient estimates, gra-dient blow up

中图分类号: 

  • 35J60