Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ ...Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.展开更多
Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are pol...Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are polynomial functions in k, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.展开更多
基金Supported by NSFC Innovation Grant(Grant No.10421101)
文摘Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.
文摘Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are polynomial functions in k, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.
