The paper is devoted to the homogenization of elliptic systems in divergence form.We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1,γdomain when the coefficients are Dini continuous,i...The paper is devoted to the homogenization of elliptic systems in divergence form.We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1,γdomain when the coefficients are Dini continuous,inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives.The results extend Avellaneda and Lin’s work[Comm.Pure Appl.Math.,40:803-847(1987)],where Holder continuity is the main assumption on smoothness of the data.展开更多
In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant n...In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.展开更多
基金Supported in part by the National Natural Science Foundation of China(No.12071365 and 12001419)。
文摘The paper is devoted to the homogenization of elliptic systems in divergence form.We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1,γdomain when the coefficients are Dini continuous,inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives.The results extend Avellaneda and Lin’s work[Comm.Pure Appl.Math.,40:803-847(1987)],where Holder continuity is the main assumption on smoothness of the data.
基金supported by the National Natural Science Foundation of China under grant No.12271108Shanghai Municipal Science and Technology Commission under grant No.22WZ2501900Innovation Program of Shanghai Municipal Education Commission
文摘In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.
