Let L be the infinitesimal generator of an analytic semigroup on L;（R;）with pointwise upper bounds on heat kernel,and denote by L;the fractional integrals of L.For a BMO function b（x）,we show a weak type Llog L es...Let L be the infinitesimal generator of an analytic semigroup on L;（R;）with pointwise upper bounds on heat kernel,and denote by L;the fractional integrals of L.For a BMO function b（x）,we show a weak type Llog L estimate of the commutators [b,L;]（f）(x) = b（x）L;（f）(x)-L;（bf）(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.展开更多
This paper considers linearized BCL system with viscosity which is firstly derived by J. L. Bona, T. Colin and D. Lannes for the study of motion of water waves. Ldecay estimate is got by means of Fourier analysis and ...This paper considers linearized BCL system with viscosity which is firstly derived by J. L. Bona, T. Colin and D. Lannes for the study of motion of water waves. Ldecay estimate is got by means of Fourier analysis and frequency decomposition. This result plays key role in studying the global well-posedness of corresponding nonlinear system.展开更多
In this paper, by using holomorphic support f unction of strictly pseudoconvex domain on Stein manifolds and the kernel define d by DEMAILY J P and Laurent Thiebaut, we construct two integral operators T q and S q whi...In this paper, by using holomorphic support f unction of strictly pseudoconvex domain on Stein manifolds and the kernel define d by DEMAILY J P and Laurent Thiebaut, we construct two integral operators T q and S q which are both belong to C s+α p,q-1 (D) and ob tain integral representation of the solution of (p,q)-form b-equation on the boundary of pseudoconvex domain in Stein manifolds and the L s p,q extimates for the solution.展开更多
Through the estimates for Greens function of linearized system, we obtain L p estimates of the asymptotic behavior of solutions of the Cauchy problem for the Navier Stokes systems of compressible flow in seve...Through the estimates for Greens function of linearized system, we obtain L p estimates of the asymptotic behavior of solutions of the Cauchy problem for the Navier Stokes systems of compressible flow in several space dimensions.展开更多
In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability...In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.展开更多
We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)...We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.展开更多
The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational eval...The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.展开更多
In this paper, a semilinear elliptic-parabolic PDE system which arises in a two dimensional groundwater flow problem is studied. Existence and uniqueness results are established via the L￣p - L￣q a priori estimates ...In this paper, a semilinear elliptic-parabolic PDE system which arises in a two dimensional groundwater flow problem is studied. Existence and uniqueness results are established via the L￣p - L￣q a priori estimates and the inverse function theorem.展开更多
In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defin...In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.展开更多
The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The p...The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in 12 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.展开更多
In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second ...In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？展开更多
We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advanta...We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.展开更多
This paper investigates L∞--estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state...This paper investigates L∞--estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. The authors derive L∞--estimates for the mixed finite element approximation of nonlinear optimal control problems. Finally, the numerical examples are given.展开更多
We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necess...We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.展开更多
In this paper,we solve the optimal constant problem in the setting of Ohsawa’s generalized L2extension theorem.As applications,we prove a conjecture of Ohsawa and the extended Suita conjecture,we also establish some ...In this paper,we solve the optimal constant problem in the setting of Ohsawa’s generalized L2extension theorem.As applications,we prove a conjecture of Ohsawa and the extended Suita conjecture,we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces.展开更多
The main purpose of this paper is to generalize the celebrated L^2 extension theorem of Ohsawa and Takegoshi in several directions: The holomorphic sections to extend are taken in a possibly singular hermitian line bu...The main purpose of this paper is to generalize the celebrated L^2 extension theorem of Ohsawa and Takegoshi in several directions: The holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K¨ahler and holomorphically convex, but not necessarily compact.展开更多
In this paper, we give a survey of various regularization theorems of quasi-plurisubharmonic functions on complex manifolds, and we also discuss the ideas used in their proofs. Moreover, we will present their applicat...In this paper, we give a survey of various regularization theorems of quasi-plurisubharmonic functions on complex manifolds, and we also discuss the ideas used in their proofs. Moreover, we will present their applications in our studies on extension problems.展开更多
For the three-dimensional compressible multicomponent displacement problem we put forward the modified method of characteristics with finite element operator-splitting procedures and make use of operator-splitting,cha...For the three-dimensional compressible multicomponent displacement problem we put forward the modified method of characteristics with finite element operator-splitting procedures and make use of operator-splitting,characteristic method,calculus of variations,energy method,negative norm estimate,two kinds of test functions and the theory of prior estimates and techniques.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.These methods have been successfully used in oil-gas resources estimation,enhanced oil recovery simulation and seawater intrusion numerical simulation.展开更多
We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous probl...We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.展开更多
基金The Science and Technology Research(Z2014057)of Higher Education in Hebei Provincethe Doctoral Foundation(L2015B05)of Hebei Normal Universitythe NSF(A2015403040)of Hebei Province
文摘Let L be the infinitesimal generator of an analytic semigroup on L;（R;）with pointwise upper bounds on heat kernel,and denote by L;the fractional integrals of L.For a BMO function b（x）,we show a weak type Llog L estimate of the commutators [b,L;]（f）(x) = b（x）L;（f）(x)-L;（bf）(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.
基金Supported by the National Natural Science Foundation of China(11571092)
文摘This paper considers linearized BCL system with viscosity which is firstly derived by J. L. Bona, T. Colin and D. Lannes for the study of motion of water waves. Ldecay estimate is got by means of Fourier analysis and frequency decomposition. This result plays key role in studying the global well-posedness of corresponding nonlinear system.
文摘In this paper, by using holomorphic support f unction of strictly pseudoconvex domain on Stein manifolds and the kernel define d by DEMAILY J P and Laurent Thiebaut, we construct two integral operators T q and S q which are both belong to C s+α p,q-1 (D) and ob tain integral representation of the solution of (p,q)-form b-equation on the boundary of pseudoconvex domain in Stein manifolds and the L s p,q extimates for the solution.
文摘Through the estimates for Greens function of linearized system, we obtain L p estimates of the asymptotic behavior of solutions of the Cauchy problem for the Navier Stokes systems of compressible flow in several space dimensions.
基金supported by NSFC(11341002)NSFC(11171104,10871066)+1 种基金the Construct Program of the Key Discipline in Hunansupported in part by US National Science Foundation under Grant DMS-1115530
文摘In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.
基金Supported by the National Natural Science Foundation of China(Grant No.11901131)the University-Level Research Fund Project in Guizhou University of Finance and Economics(Grant No.2019XYB08)。
文摘We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.
基金supported by the Major State Basic Research Development Program of China(No.G19990328)the National Key Technologies R&D Program of China (No.20050200069)+1 种基金the National Natural Science Foundation of China (Nos.10771124 and 10372052)the Ph. D. Pro-grams Foundation of Ministry of Education of China (No.20030422047)
文摘The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.
文摘In this paper, a semilinear elliptic-parabolic PDE system which arises in a two dimensional groundwater flow problem is studied. Existence and uniqueness results are established via the L￣p - L￣q a priori estimates and the inverse function theorem.
基金This work was partially supported by the National Natural Science Foundation of China(No.11871009).
文摘In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.
基金Supported by the National Natural Science Foundation of China(11101124 and 11271231)Natural Science Foundation of Shandong Province(ZR2016AM08)National Tackling Key Problems Program(2011ZX05052,2011ZX05011-004)
文摘The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in 12 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.
基金supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905)Shang hai Leading Academic Discipline Project (Grant No. J50101)+3 种基金Key Disciplines of Shanghai Municipality (GrantNo. S30104)Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700)National Natural Science Foundation of China (Grant No. 60532080)the Key Project of Chinese Ministryof Education (Grant No. 306017)
文摘In this paper we obtain local Lp estimates for the parabolic polyharmonic equations by a straightforward approach.
文摘In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？
文摘We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.
基金supported by the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China under Grant No.10971074China Postdoctoral Science Foundation under Grant No.2011M500968
文摘This paper investigates L∞--estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. The authors derive L∞--estimates for the mixed finite element approximation of nonlinear optimal control problems. Finally, the numerical examples are given.
文摘We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.
基金supported by National Natural Science Foundation of China (Grant No. 11031008)
文摘In this paper,we solve the optimal constant problem in the setting of Ohsawa’s generalized L2extension theorem.As applications,we prove a conjecture of Ohsawa and the extended Suita conjecture,we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces.
基金supported by the Agence Nationale de la Recherche grant“Convergence de Gromov-Hausdorff en géeométrie khlérienne”the European Research Council project“Algebraic and Khler Geometry”(Grant No.670846)from September 2015+1 种基金the Japan Society for the Promotion of Science Grant-inAid for Young Scientists(B)(Grant No.25800051)the Japan Society for the Promotion of Science Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers
文摘The main purpose of this paper is to generalize the celebrated L^2 extension theorem of Ohsawa and Takegoshi in several directions: The holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K¨ahler and holomorphically convex, but not necessarily compact.
基金supported by National Natural Science Foundation of China(Grant Nos.11688101 and 11431013)Langfeng Zhu was partially supported by National Natural Science Foundation of China(Grant Nos.11201347 and 11671306)
文摘In this paper, we give a survey of various regularization theorems of quasi-plurisubharmonic functions on complex manifolds, and we also discuss the ideas used in their proofs. Moreover, we will present their applications in our studies on extension problems.
基金This research is supported by the Major State Research Program of China(Grant No.19990328),the National Natural Sciences Foundation of China(Grant Nos.19871051 and 19972039),the National Tackling Key Problems Program and the Doctorate Foundation of the S
文摘For the three-dimensional compressible multicomponent displacement problem we put forward the modified method of characteristics with finite element operator-splitting procedures and make use of operator-splitting,characteristic method,calculus of variations,energy method,negative norm estimate,two kinds of test functions and the theory of prior estimates and techniques.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.These methods have been successfully used in oil-gas resources estimation,enhanced oil recovery simulation and seawater intrusion numerical simulation.
基金supported in part by NSF Awards 0715146,0821816,0915220 and 0822283(CTBP)NIHAward P41RR08605-16(NBCR),DOD/DTRA Award HDTRA-09-1-0036+1 种基金CTBP,NBCR,NSF and NIHsupported in part by NIH,NSF,HHMI,CTBP and NBCR.The third,fourth and fifth authors were supported in part by NSF Award 0715146,CTBP,NBCR and HHMI.
文摘We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
