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 paper,we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary....In this paper,we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary.If the domain satisfies C1,Dinicondition at a boundary point,and the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition,then the solution is Lipschitz continuous at this point.Furthermore,we generalize this result to Reifenberg C1,Dinidomains.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[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution.Their experimental results show that the detail of the restored images cannot be r...In[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution.Their experimental results show that the detail of the restored images cannot be recovered.In this paper,we consider images in Lipschitz spaces,and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind deconvolution.Our experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.展开更多
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The third author was partially supported by NSFC(Grant Nos.11771285 and 12031012)。
文摘In this paper,we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary.If the domain satisfies C1,Dinicondition at a boundary point,and the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition,then the solution is Lipschitz continuous at this point.Furthermore,we generalize this result to Reifenberg C1,Dinidomains.
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.
基金This research is supported in part by RGC 7046/03P,7035/04P,7035/05P and HKBU FRGs.
文摘In[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution.Their experimental results show that the detail of the restored images cannot be recovered.In this paper,we consider images in Lipschitz spaces,and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind deconvolution.Our experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.