Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

---

Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...     

双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.     

关于数列{2~n}

易知等比数列{2~n}前n项的和为S_n=a_(n+1)-2。对于这个关系式,我们有三点联想。 (一)简便求和。若a_5=32,则S_4=30。 (二)判定由关系式a_(n+1)=rS_n+S给出的数列是否为等比数列。事实上a_(n+1)=rS_n+S (1) a~n=rS_(n-1)+S (2) (1)-(2)得a_(n+1)-a_n=r(s_n-S_(n-1))=ra_n a_(n+1)/a_n=r+ 因此,r≠-1,a_1=S,{a_n}为等比数列。 (三)等比数列前n项和公式的新法推导。     

ALTERNATING BAND CRANK-NICOLSON METHOD FOR δu／δt=δ^2u／δx^2＋δ^2u／δy^2

the Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [1],and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2].The methods have the advantages of parallel computing,stability and good accuracy.In this paper for the two-dimensional diffusion equation,the net region is divided into bands,a special kind of block.This method is called the alternating Band Crank-Nicolson method.     

偏微分方程△^2u—s△u＋k^2u=0边值问题的MRM边界变分方程

导出边值问题△^2u-s△u k^2u=0；x∈Ω∪Ω‘∪→R^2;u/Γ=u0;δu/δn/Γ=g0的定解问题，MRM边界变分方程，全平面解的表达式，从中可以以看出，MRM边界变分方程中只包含弱奇异积分核，并且自动消除了原第一，二MRM边界积分方程中出现的强奇异积分核，问题解的表达式后并不加任何多项式，因而也不需要引入Lagrange乘子求解该项，这给边界元数值求解过程带来极大的方便，数值分析结果表明该方法具有明显优势。     

Gradient Estimates and Harnack Inequalities for Positive Solutions of $$\mathfrak{L}u=\frac{\partial u}{\partial t}$$ L u = ∂ u ∂ t on Self-shrinkers

In this paper, we investigate the positive solutions of Lu=∂u/∂t on self-shrinkers, then get some gradient estimates and Harnack inequalities for the positive solutions.     

具临界指数椭圆方程-Δu=λ_κu |u|~(2~*-2)u f(x,u)非平凡多解存在性

本文利用山路引理以及P．L．Lions的集中紧性原理,给出了具临界指数2*且涉及任意特征值λk 的Dirichlet问题-△u=λku |u|2*-2u f(x,u)一对非平凡解的存在性定理,其中次临界扰动项 f(x,t)可以是关于变量t的非线性项．     

关于同伦元素 $\alpha_{1}\beta_{1}\beta_{2}\gamma_{s}$

设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s 相似文献   

具临界指数椭圆方程-△u=λku+|u|2*-2u+f(x,u)非平凡多解存在性

本文利用山路引理以及P.L.Lions的集中紧性原理,给出了具临界指数2*且涉及任意特征值λk的Dirichlet问题-△u=λku+|u|2*-2u+f(x,u)一对非平凡解的存在性定理,其中次临界扰动项f(x,t)可以是关于变量t的非线性项.     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

The Brownian snake and solutions of Δu=u 2 in a domain

Summary We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation u=u ² in a domain of ^d . In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation u=u ². The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.     

Linear Codes over \mathbb{F}_{q}[u]/(u^s) with Respect to the Rosenbloom–Tsfasman Metric

We investigate the structure of codes over rings with respect to the Rosenbloom-Tsfasman (RT) metric. We define a standard form generator matrix and show how we can determine the minimum distance of a code by taking advantage of its standard form. We define MDR (maximum distance rank) codes with respect to this metric and give the weights of the codewords of an MDR code. We explore the structure of cyclic codes over and show that all cyclic codes over rings are MDR. We propose a decoding algorithm for linear codes over these rings with respect to the RT metric. AMS Classification: 94B05, 94B60     

r模同态的{1}-逆与{2}-逆

本文探讨模同态广义逆在环模理论中的应用．利用模同态的{1}-逆与{2}-逆，分别给出了一类环及一类重要模的特征刻画．     

关于环$Z_4[u]/\langle u2-2\rangle$上的斜多元循环码

本文探索了环$R=Z_4[u]/\langle u2-2\rangle$ 上的几类斜多元循环码和多元循环码. 首先得到了环$R$上$(1,2u)$-多元循环码的生成多项式. 其次由定义的Gray映射得到了环$R$上$(1,2u)$- 多元循环码的Gray像是$Z_4$上的循环码或指数为2的逆循环码. 最后, 通过环$R$上$(1,2u)$- 多元循环码的一些例子来展示本文的主要结果.     

关于方程组{x^2＋y^2＋z^2=u ax＋by＋cz=v及其推广形式的解与解法的研究

利用柯西不等式及球面与平面的位置关系，对方程组{x^2＋y^2＋z^2=u ax＋by＋cz=v（其中各参数均为实数，且u〉0,a,b,c全不为零）及其推广形式的解的情况及解法进行探讨。利用引理重点讨论椭球面方程与平面方程的解法与解法与解     

Surfaces with $${K^2=2\mathcal{X}-2}$$ and pg ≥ 5

This note describes minimal surfaces S of general type satisfying p _g  ≥ 5 and K ² = 2p _g . For p _g  ≥ 8 the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if p _g  ≥ 13, S has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for p _g  ≤ 12 there are two other classes of such surfaces with non birational canonical map.     

关于方程Δu-q■u=(■)~2u/(■)t~2的柯西问题解的公式

<正> 本文用逐步逼近法来解如下的柯西问题:这里q(x),f(x)是定义在全空间中的连续函数,并具有二阶有界导数。     

方程~2u/x~1x~1 ~2u/x~2x~2=f(u)的Bcklund变换

本文利用Wahlquist-Estabrook过程(WEP)研究了方程(?)~2u/(?)x~1(?)x~1 (?)~2u/(?)x~2(?)x~2=f(u)(这里f是任意函数)的B(?)cklund变换.我们发现该方程存在B(?)cklund变换的充分条件是d~2f/du~2=λf.我们所得到的结果的一个特殊情况就是Leibbrandt的结论.