共查询到20条相似文献,搜索用时 62 毫秒
1.
Hu Ke 《数学年刊B辑(英文版)》1980,1(34):421-427
Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l}
{\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m},
\end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m.
Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l}
\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3,
\end{array}\], where \[\begin{array}{l}
{F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}).
\end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun.
Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l}
P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2},
\end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing. 相似文献
2.
A. P. Oskolkov 《Journal of Mathematical Sciences》1996,79(3):1129-1145
Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem 9 $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ 10 $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ . We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles. 相似文献
3.
Minbo Yang Yuanhong Wei Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2014,65(1):41-68
In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type $$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$ Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter ${\varepsilon}$ is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents. 相似文献
4.
Let $I_{\alpha,\vec{b}}$ be the multilinear commutators of the fractional integrals $I_{\alpha}$ with the symbol $\vec{b}=(b_1, \cdots,b_k )$. We show that the constant of borderline weighted estimates for $I_{\alpha}$ is $\frac{1}{{\varepsilon}}$, and for $I_{\alpha,{\vec{b}}}$ is $\frac{1}{{\varepsilon}^{k+1}}$ with each $b_i$ belongs to the Orlicz space $Osc_{\exp L^{s_i}}$. 相似文献
5.
Weiyang Chen & Xiaoli Chen 《数学研究》2014,47(2):208-220
In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha
1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$ 相似文献
6.
刘桥 《数学年刊A辑(中文版)》2014,35(5):591-612
考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的. 相似文献
7.
A. A. Kotsiolis 《Journal of Mathematical Sciences》1997,83(2):233-243
In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following
two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:
and the following two dissipative ε-approximations for the equations of motion of the Kelvin-Voight fluids:
satisfying the free surface conditions on the boundary ϖΩ of a domain Ω⊂R3:
.
The free term f(x, t) in systems (1)–(4) is assumed to be t-periodic with period T. It is shown that as ε→0, the classical
t-periodic solutions (with period T) of Eqs. (1)–(4) satisfying the free surface conditions (5) converge to the classicat
t-periodic solutions (with period T) of the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya and to the
equations of motion of the Kelvin-Voight fruids, respectively, satisfying the boundary condition (5).
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 109–124.
Translated by N. S. Zabavnikova. 相似文献
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8.
In this paper we study the first and tiie third boundary value problems for the elliptic equation
\[\begin{array}{l}
\varepsilon \left( {\sum\limits_{i,j = 1}^m {{d_{i,j}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{i = 1}^m {{d_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + d(x)u} } } \right) + \sum\limits_{i = 1}^m {{a_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + b(x) + c} \ = f(x),x \in G(0 < \varepsilon \le 1),
\end{array}\]
as the degenerated operator bas singular points, where
\[\sum\limits_{i,j = 1}^m {{d_{i,j}}(x){\xi _i}{\xi _j}} \ge {\delta _0}\sum\limits_{i = 1}^m {\xi _i^2} ,({\delta _0} > 0,x \in G).\]
The uniformly valid asymptotic solutions of boundary value problems have been
obtained under the condition of
\[\sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} > 0,or} \sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} < 0} ,\]
where \(n = ({n_1}(x),{n_2}(x), \cdots ,{n_m}(x))\) is the interior normal to \({\partial G}\). 相似文献
9.
N. B. Maslova 《Journal of Mathematical Sciences》1989,45(3):1141-1153
One investigates the Cauchy problem for the nonlinear Boltzmann equation
相似文献
10.
B. D. Boyanov 《Mathematical Notes》1973,14(1):551-556
The author considers a class F of analytic functions real in the interval [-1, 1] and bounded in the unit circle. As an estimate of the optimal quadrature error R(n) over the class F it is shown that $$_e - \left( {2\sqrt 2 + \frac{1}{{\sqrt 2 }}} \right)\pi \sqrt n \leqslant R(n) \leqslant e^{ - \frac{\pi }{{\sqrt 2 }}n} .$$ With the additional condition that \(\mathop {max}\limits_{x \in [ - 1,1]}\) ¦f(x)¦?B, an estimate is obtained for the ?-entropy H?(F): $$\frac{8}{{27}}\frac{{(1n2)^2 }}{{\pi ^2 }} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{H_\varepsilon (F)}}{{\left( {\log \frac{1}{\varepsilon }} \right)^3 }} \leqslant \frac{2}{{\pi ^2 }}(1n2)^2 .$$ 相似文献
11.
Classification of positive solutions to an integral system with the poly-harmonic extension operator
In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system. 相似文献
12.
Sadek Gala 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):181-194
In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in Morrey–Campanato
spaces. It is proved that if the velocity field satisfies
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