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
In this paper we first give an a priori estimate of maximum modulus ofsolutions for a class of systems of diagonally degenerate elliptic equations in the case of p > 2.
In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the measure.Moreover,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤...In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the measure.Moreover,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤c(p,t)(Pt|f|p)^(1/p),p>1,t>0 is obtained for the associated nonlinear semigroup P¯t.As an application of the Harnack inequality,we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition.Finally,an example is presented.All of the above results extend the existing ones in the linear expectation setting.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In this paper we first give an a priori estimate of maximum modulus ofsolutions for a class of systems of diagonally degenerate elliptic equations in the case of p > 2.
基金supported by National Natural Science Foundation of China(Grant No.11801406).
文摘In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the measure.Moreover,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤c(p,t)(Pt|f|p)^(1/p),p>1,t>0 is obtained for the associated nonlinear semigroup P¯t.As an application of the Harnack inequality,we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition.Finally,an example is presented.All of the above results extend the existing ones in the linear expectation setting.
