In this paper, we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem, the Taylor formula and a priori estimates for the derivatives of the Ne...In this paper, we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem, the Taylor formula and a priori estimates for the derivatives of the Newton potential.展开更多
Global Schauder estimates for the initial-value parabolic problem of the hi-harmonic type are proved, and the existence and uniqueness of the solutions in the suitable space are obtained. Similarly to the second-order...Global Schauder estimates for the initial-value parabolic problem of the hi-harmonic type are proved, and the existence and uniqueness of the solutions in the suitable space are obtained. Similarly to the second-order case, first a formal expression of solutions by the Fourier transform is obtained, and then the regularity, uniqueness and existence of solutions using the potential theory and approximation argument are got. out approach is simple and straightforward.展开更多
In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.
We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas ...We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.展开更多
In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of...In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.展开更多
We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-...We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-projective manifolds.The main tools are the solution formula for second-order ordinary differential equations(ODEs)with constant coefficients and spectral theory for the Laplacian operator on a closed manifold.展开更多
The author establishes the long-time existence and convergence results of the mean curvature flow of entire Lagrangian graphs in the pseudo-Euclidean space,which is related to the logarithmic Monge-Ampere flow.
基金supported by the NSFC(11201486,11326153)supported by"the Fundamental Research Funds for the Central Universities(31541411213)"
文摘In this paper, we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem, the Taylor formula and a priori estimates for the derivatives of the Newton potential.
基金Project supported by the Major State Basic Research Development Program of China (973 Program) (No.2006CB705700)the National Natural Science Foundation of China (No.60532080)the Key Project of Chinese Ministry of Education (No.306017)
文摘Global Schauder estimates for the initial-value parabolic problem of the hi-harmonic type are proved, and the existence and uniqueness of the solutions in the suitable space are obtained. Similarly to the second-order case, first a formal expression of solutions by the Fourier transform is obtained, and then the regularity, uniqueness and existence of solutions using the potential theory and approximation argument are got. out approach is simple and straightforward.
基金Project supported by the Australian Research Council and the National Natural Science Foundation of China (No.10428103).
文摘In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.
基金supported by Simons Foundation(Grant No.580911(Stinga))Ministerio de Economía y Competitividad/Fondo Europeo de Desarrollo Regional from Government of Spain(Grant No.MTM2015-66157-C2-1-P(Torrea))。
文摘We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.
基金the Tianyuan Fund for Mathematics(No.10526022)the 985 program of Jilin University
文摘In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.
基金supported by National Natural Science Foundation of China(Grant Nos.11331001 and 11871265)the Hwa Ying Foundation for its financial support and thanks Professor Jian Song for his invitation。
文摘We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-projective manifolds.The main tools are the solution formula for second-order ordinary differential equations(ODEs)with constant coefficients and spectral theory for the Laplacian operator on a closed manifold.
文摘The author establishes the long-time existence and convergence results of the mean curvature flow of entire Lagrangian graphs in the pseudo-Euclidean space,which is related to the logarithmic Monge-Ampere flow.
