In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the bou...In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the boundary value is . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation in. We propose two different conditions for the right hand side and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.展开更多
We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and singular parabolic inhomogeneous equation of the form ut-△ ∞N u=f, wher...We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and singular parabolic inhomogeneous equation of the form ut-△ ∞N u=f, where An denotes the so-called normalized infinity Laplacian given by △∞ Nu=1/｜Du｜2〈D2uDu,Du〉.展开更多
In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N...In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N)_(∞)u+β|D u|,which arises from the random game named biased tug-of-war.By studying directly theβ−biased infinity Laplacian equation,we construct the appropriate exponential cones as barrier functions to establish a key estimate.Based on this estimate,we obtain the Harnack inequality,Hopf boundary point lemma,Lipschitz estimate and the Liouville property etc.展开更多
In this paper, we study the viscosity solutions of the Neumann problem in a bounded C<sup>2</sup> domain Ω, where Δ<sup>N</sup>∞</sub> is called the normalized infinity Laplacian. The ...In this paper, we study the viscosity solutions of the Neumann problem in a bounded C<sup>2</sup> domain Ω, where Δ<sup>N</sup>∞</sub> is called the normalized infinity Laplacian. The normalized infinity Laplacian was first studied by Peres, Shramm, Sheffield and Wilson from the point of randomized theory named tug-of-war, which has wide applications in optimal mass transportation, financial option price problems, digital image processing, physical engineering, etc. We give the Lipschitz regularity of the viscosity solutions of the Neumann problem. The method we adopt is to choose suitable auxiliary functions as barrier functions and combine the perturbation method and viscosity solutions theory. .展开更多
In this article, we prove that viscosity solutions of the parabolic inhomogeneous equationsn＋p/put-△p^Nu=f（x,t）can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞...In this article, we prove that viscosity solutions of the parabolic inhomogeneous equationsn＋p/put-△p^Nu=f（x,t）can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.展开更多
文摘In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is and the boundary value is . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation in. We propose two different conditions for the right hand side and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.
基金Supported by National Natural Science Foundation of China(Grant Nos.11071119 and 11171153)
文摘We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and singular parabolic inhomogeneous equation of the form ut-△ ∞N u=f, where An denotes the so-called normalized infinity Laplacian given by △∞ Nu=1/｜Du｜2〈D2uDu,Du〉.
基金the Fundamental Research Funds for the Central Universities(Grant No.30919013235)National Natural Science Foundation of China(Nos.11501292 and 11501293).
文摘In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N)_(∞)u+β|D u|,which arises from the random game named biased tug-of-war.By studying directly theβ−biased infinity Laplacian equation,we construct the appropriate exponential cones as barrier functions to establish a key estimate.Based on this estimate,we obtain the Harnack inequality,Hopf boundary point lemma,Lipschitz estimate and the Liouville property etc.
文摘In this paper, we study the viscosity solutions of the Neumann problem in a bounded C<sup>2</sup> domain Ω, where Δ<sup>N</sup>∞</sub> is called the normalized infinity Laplacian. The normalized infinity Laplacian was first studied by Peres, Shramm, Sheffield and Wilson from the point of randomized theory named tug-of-war, which has wide applications in optimal mass transportation, financial option price problems, digital image processing, physical engineering, etc. We give the Lipschitz regularity of the viscosity solutions of the Neumann problem. The method we adopt is to choose suitable auxiliary functions as barrier functions and combine the perturbation method and viscosity solutions theory. .
基金supported by the National Natural Science Foundation of China(11071119,11171153)
文摘In this article, we prove that viscosity solutions of the parabolic inhomogeneous equationsn＋p/put-△p^Nu=f（x,t）can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.
