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Oblique derivative problem for general Chaplygin-Rassias equations

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Science china Mathematics 2008年第1期51卷 5-36页

作者： WEN GuoChun LMAM, school of mathematical sciences, peking university, beijing 100871, china LMAM School of Mathematical Sciences Peking University Beijing 100871 China

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 < μ < 1) is a curve with the end points z = ?1, 1. L 1, L 2, L 3, L 4 are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y < 0}) passing through the points z = x + iy = ?1, 0, 0, 1 respectively. And the boundary condition possesses the form $$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H

关键词： oblique derivative problem equations of mixed type nonsmooth degenerate line 35J70 35L80 35N99

Global well-posedness for the 2D critical dissipative quasi-geostrophic equation

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Science china Mathematics 2007年第4期50卷 485-494页

作者： Zhi-fei ZHANG school of mathematical sciences, peking university, beijing 100871, china school of mathematical sciences Peking University Beijing 100871 China

We study the initial value problem for the 2D critical dissipative quasi-geostrophic equation. We prove the global existence for small data in the scale invariant Besov spaces ? p,1 2/p , 1 ? p ? ∞. In particular, for p = ∞, our result does not impose any regularity on the initial data. Our proofs are based on an exponential decay estimate of the semigroup $e^{{ - t\kappa ( - \Delta )}^{\alpha}} $ and the use of space-time Besov spaces.

关键词： quasi-geostrophic equation Besov space Littlewood-Paley decomposition well-posedness 35Q35

Maslov-type index and brake orbits in nonlinear Hamiltonian systems

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Science china Mathematics 2007年第6期50卷 761-772页

作者： Duan-zhi ZHANG school of mathematical sciences, peking university, beijing 100871, china 1. school of mathematical sciences Peking University Beijing 100871 China

In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian s... 详细信息

关键词： brake orbit Maslov-type index relative Morse index dual variational method 58E05 70H05 34C25

Finite Groups in Which Sylow Normalizers Admit Supplements

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Acta Mathematica Sinica,English Series 2007年第3期23卷 417-422页

作者： Tian Ze LI school of mathematical sciences,peking university,beijing 100871,P.R.china school of mathematical sciences Peking University Beijing 100871 P. R.China

For a group G and a subgroup M of G,we say that a subgroup A of G is a supplement toM in G,if G=*** prove that a finite group in which every Sylow normalizer admits a nilpotentsupplement is *** particular we confirm a... 详细信息

关键词： Sylow normalizer Supplement Wreath product

Uniqueness and local stability for the inverse scattering problem of determining the cavity

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Science china Mathematics 2005年第8期48卷 1113-1123页

作者： FENG Lixin & MA Fuming LMAM, school of mathematical sciences, peking university, beijing 100871, china school of mathematical sciences, Heilongjiang university, Harbin 150080, china school of mathematical sciences, Jilin university, Changchun 130012, china 1. LMAM School of Mathematical Sciences Peking University 100871 Beijing China 2. school of mathematical sciences Heilongjiang University 150080 Harbin China 3. school of mathematical sciences Jilin University 130012 Changchun China

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.

关键词： uniqueness, stability, Maxwell's equations, inverse scattering problem.

The lifting scheme of 4-channel orthogonal wavelet transforms

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Progress in Natural Science:Materials International 2006年第1期16卷 100-104页

作者： PENG Lizhong and CHU Xiaoyong (LMAM, school of mathematical sciences, peking university, beijing 100871, china) [a]peking Universiy LMAM School of Mathematical Sciences Beijing China

The 4-channel smooth wavelets with linear phase and orthogonality are designed from the 2-channel orthogonal wavelets with high transfer vanishing moments. Reversely, for simple lifting scheme of such 4-channel orthogonal wavelet transforms, a new 2-channel orthogonal wavelet associated with this 4-channel wavelet is constructed. The new 2-channel wavelet has at least the same number of vanishing moments as the associated 4-channel one. Finally, by combining the two such 2-channel wavelet systems, the lifting scheme of 4-channel orthogonal wavelet transform, which has simple structure and is easy to apply, is presented.

关键词： lifting scheme 4-channel orthogonal wavelet 2-channel orthogonal wavelet vanishing moment transfer vanishing moment linear phase.

OBLIQUE DERIVATIVE PROBLEMS FOR SECOND ORDER NONLINEAR MIXED EQUATIONS WITH DEGENERATE LINE

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Acta Mathematica Scientia 2008年第3期28卷 604-612页

作者：闻国椿 school of mathematical sciences Peking University

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.

关键词： Oblique derivative problems nonlinear mixed equations parabolic degeneracy

r-matrix and algebraic-geometric symplectic map

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Chinese Science Bulletin 1999年第2期44卷 114-118页

作者： QIAO ZhijunInstitute of Mathematics and school of mathematical sciences, peking university, beijing 100871, china Institute of Mathematics and school of mathematical sciences Peking University Beijing China

A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical(i. e. constant) r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution(here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix, r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.

关键词： symplectic map r-matrix algebraic-geometric solution.

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

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Acta Mathematica Scientia 2007年第3期27卷 663-672页

作者：闻国椿 school of mathematical sciences Peking University

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.

关键词： Boundary value problems nonlinear elliptic equations parabolic degeneracy