Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

Sobolev空间的K泛函

<正> Devore和Schere在[1]中用样条函数的方法得出了Sobolev空间(w_1~h(R~n),w_∞~k(R~n))的K泛函等价关系:     

Morrey–Sobolev Extension Domains

We show that every uniform domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 2\) is a Morrey–Sobolev \({\mathscr {W}}^{1,\,p}\)-extension domain for all \(p\in [1,\,n)\), and moreover, that this result is essentially the best possible for each \(p\in [1,\,n)\) in the sense that, given a simply connected planar domain or a domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 3\) that is quasiconformal equivalent to a uniform domain, if it is a \({\mathscr {W}}^{1,\,p} \)-extension domain, then it must be uniform.     

Sharp Hardy–Sobolev inequalities

Let Ω be a smooth bounded domain in ${\mathbb{R}}^N$, $N?3$. We show that Hardy's inequality involving the distance to the boundary, with best constant ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$), may still be improved by adding a multiple of the critical Sobolev norm. To cite this article: S. Filippas et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Fractional Sobolev,Moser–Trudinger,Morrey–Sobolev inequalities under Lorentz norms

We consider the Sobolev type inequalities under Lorentz norms on bounded open domains for fractional derivatives (−∆) ^s/2 in the following three cases: n > ps, n = ps, and n < ps, whence establishing the weak type Sobolev inequalities, Moser–Trudinger and Morrey–Sobolev inequalities for fractional derivatives in Lorentz norms. Applying these inequalities, we obtain the trace forms of six related functional inequalities. Bibliography: 44 titles.     

非负Ricci曲率Riemann流形上的Sobolev不等式及Sobolev嵌入定理

设Ｍ是具有非负Ｒｉｃｃｉ曲率的完备Ｒｉｅｍａｎｎ流形，本文证明Ｍ上Ｓｏｂｏｌｅｖ不等式‖ｆ‖ｑ≤Ｃｎ，ｐ，ｑ（１≤Ｐ，ｑ＜∞）对一切（Ｍ）成立的充要条件是对一切ｘ∈Ｍ，Ｖｘ（ｒ）＝Ｖｏｌ（Ｂｘ（ｒ））≥且，而Ｍ上较弱的Ｓｏｂｏｌｅｖ不等式‖ｆ‖ｑ≤Ｃｎ‖Ｆ‖ｐ）（１＜ｐ＜ｑ＜∞）对一切ｆ∈Ｈ（Ｍ）成立的充要条件是，且最后，证明了Ｍ上ｓｏｂｏｌｅｖ嵌入定理，如果，则；如果则成立．     

Sobolev空间的实内插

本文求出了Sobolev空间L_k^∞,BMO_k与L_k^p,H_k^p(0 < p < ∞)之间的实内插空间.     

Positive Solutionsofp-LaplaceEquation Involving Critical Sobolev Exponent

ＰｏｓｉｔｉｖｅＳｏｌｕｔｉｏｎｓｏｆｐ－ＬａｐｌａｃｅＥｑｕａｔｉｏｎＩｎｖｏｌｖｉｎｇＣｒｉｔｉｃａｌＳｏｂｏｌｅｖＥｘｐｏｎｅｎｔＳｕｎＧｕｏｈｏｎｇ（孙国弘）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈ...     

优化Sobolev体及其仿射性质

Sobolev不等式是联系分析和几何的基础不等式之一,而优化Sobolev体是优化Sobolev范数的临界几何核.首先,证明优化Sobolev体的一些仿射性质.然后,运用Barthe的优化迁移方法研究了凸体的特征函数和多胞形仿射函数的优化Sobolev体.     

Sobolev方程的等参有限元法

用等参有限元去逼近曲边区域可以使误差阶不受损失,达到和凸多边形区域上同样的收敛阶.本文研究Sobolev方程的等参有限元方法,在半离散和向后欧拉全离散格式下,分别讨论真解和有限元解之间的误差估计,最后用数值算例进一步验证了理论结果.     

Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents

We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where ${ \mu \ge 0, 0 < s < 2, 0 \in \partial \Omega}$ and Ω is a bounded domain in R ^N . We prove that if ${N \ge 7, a(0) > 0}$ and all the principle curvatures of ?Ω at 0 are negative, then the above problem has infinitely many solutions.     

Sharp convex Lorentz–Sobolev inequalities

New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L ^p Minkowski problem. New L ^p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

对称型的对数Sobolev不等式

使用新型Cheeger常数 ,给出了关于一般对称型的对数Sobolev常数的一些估计 .这些估计在某种意义上是精确的     

Sobolev圆盘代数上的乘法算子

研究一类由单位圆盘D上的Sobolev空间W^2,2(D)中的解析函数构成的代数, 称之为Sobolev圆盘代数, 给出了其上的有界线性乘法算子M_f的基本性质, 刻画了乘法算子M_f的换位子代数, 证明了A′(M_f)是交换的当且仅当M_f^*是指标为1的Cowen-Douglas算子.     

加权Sobolev空间的完备性

本文研究加权Soblev空间完备性 .我们运用Ap 权的构造性质证明 ,当 1

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