In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt dom...In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.展开更多
In this article, we borrow the idea of using Schur＇s test to characterize the compactness of composition operators on the weighted Bergman spaces in a bounded symmetricdomain Ω and verify that Cφ is compact on Lqa...In this article, we borrow the idea of using Schur＇s test to characterize the compactness of composition operators on the weighted Bergman spaces in a bounded symmetricdomain Ω and verify that Cφ is compact on Lqa（Ω,dvβ）if and only if K（φ（z）,φ（z））/K（z,z）→0 as z→ Ω under a mild condition,where K（z,w）is the Bergman kernel.展开更多
The use of finite element method leads to replacing the initial domain by an approaching domain. Under some appropriate assumptions, we prove that there exists a W1,+∞-diffeomorphism from the original domain to the a...The use of finite element method leads to replacing the initial domain by an approaching domain. Under some appropriate assumptions, we prove that there exists a W1,+∞-diffeomorphism from the original domain to the approaching domain.展开更多
Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts,including solution of partial differential equations(PDEs).We describe a solver for multiscale fully nonl...Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts,including solution of partial differential equations(PDEs).We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition,an accelerated Schwarz framework,and two-layer neural networks to approximate the boundary-to-boundarymap for the subdomains,which is the key step in the Schwarz procedure.Conventionally,the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain.By leveraging the compressibility of multiscale problems,our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map.Our method is applied to a multiscale semilinear elliptic equation and a multiscale p-Laplace equation.In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.展开更多
The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （...The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （2017）, where LP-boundedness is shown to fail when either the ＂near＂ C2 boundary regularity, or the strongC-linear convexity assumption is dropped.展开更多
This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DD...This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.展开更多
基金supported by the National Natural Science Foundation of China(11001246,11101139)Zhejiang Innovation Project(T200905)
文摘In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.
基金Supported by the National Natural Science Foundation of China (10771064)Natural Science Foundation of Zhejiang Province (Y7080197, Y6090036, Y6100219)+1 种基金Foundation of Creative Group in Colleges and Universities of Zhejiang Province (T200924)Foundation of Department of Education of Zhejiang province (20070482)
文摘In this article, we borrow the idea of using Schur＇s test to characterize the compactness of composition operators on the weighted Bergman spaces in a bounded symmetricdomain Ω and verify that Cφ is compact on Lqa（Ω,dvβ）if and only if K（φ（z）,φ（z））/K（z,z）→0 as z→ Ω under a mild condition,where K（z,w）is the Bergman kernel.
基金Partially supported by Professor Xu Yuesheng and his program "One hundred distinguished Young Scientists" Partially supported by "Programme Sino-Francais de Recherches Advancees(PRA).
文摘The use of finite element method leads to replacing the initial domain by an approaching domain. Under some appropriate assumptions, we prove that there exists a W1,+∞-diffeomorphism from the original domain to the approaching domain.
基金supported in part by National Science Foundation via grant 1934612a DOE Subcontract 8F-30039 from Argonne National Laboratory+1 种基金an AFOSR subcontract UTA20-001224 from UT-Austin.The work of SCsupported in part by NSF-DMS-1750488 and ONR-N00014-21-1-2140.
文摘Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts,including solution of partial differential equations(PDEs).We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition,an accelerated Schwarz framework,and two-layer neural networks to approximate the boundary-to-boundarymap for the subdomains,which is the key step in the Schwarz procedure.Conventionally,the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain.By leveraging the compressibility of multiscale problems,our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map.Our method is applied to a multiscale semilinear elliptic equation and a multiscale p-Laplace equation.In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.
基金supported by the National Science Foundation of USA (Grant Nos. DMS1503612 (Lanzani) and DMS-1265524 (Stein))
文摘The purpose of this paper is to complement the results by Lanzani and Stein （2017） by showing thedense definability of the Cauchy-Leray transform for the domains that give the counter-examples of Lanzani andStein （2017）, where LP-boundedness is shown to fail when either the ＂near＂ C2 boundary regularity, or the strongC-linear convexity assumption is dropped.
基金supported partly by National Key R&D Program of China(grants Nos.2019YFA0709600 and 2019YFA0709602)National Natural Science Foundation of China(grants Nos.11831016 and 12101609)the Innovation Foundation of Qian Xuesen Laboratory of Space Technology。
文摘This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.
