Minimization problem on the Hardy-Sobolev inequality
Minimization problem on the Hardy-Sobolev inequality
复制标题
Hardy-Sobolev 不等式的最小化问题
DOI：
10.1007/s00030-017-0447-9
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Hashizume Masato
中科院分区：
文献类型：
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作者：
Hashizume Masato
研究了Hardy-Sobolev型不等式的极小化问题。我们考虑奇点在有界域Ω⊂R^NΩ⊂RN内部的情形。到目前为止，带边界奇点的Hardy-Sobolev型不等式的最佳常数的可达性已经被研究过了，例如GhoSoub和Kang(Ann Inst Henri Poincare anal Non Lineaire 21(6)：767-793,2004)，Ghouse Soub和Robert(IMRP 21867：1-85,2006)，Ghouse Soub和Robert(Trans am Math Soc 361(9)：4843-4870,2009)等。根据他们的结果，∂Ω∂Ω在奇点的平均曲率影响最佳常数的可达性。与边界奇性情形不同，在内部奇性情形下，我们知道对于所有有界域μ，最佳哈迪-索波列夫常数Ω_S(Ω)，∫Ω)：=\Left{∗u|^2 dx\Bigg|u∈H_0^1(Ω)，\int_Ω|u|^2^*(S)|x|^S dx=1\Right\}μS(Ω)：=∫Ω|∇u|2 dx|u∈H 0 1(Ω)，∫Ω|u|2∗(S)|x|sdx=1)对于所有有界域ΩΩ永远得不到。我们可以看到，奇点在区域上的位置与极小元的存在有关。本文考虑嵌入H^1(Ω)↪L^2^*(Ω，|x|^-S dx)H1(Ω)↪L 2∗(S)(Ω，|x|-sdx))的有界域ΩΩ具有0∈Ω0∈Ω的最佳常数的可达性。在这个问题中，尺度不变性是不成立的，我们不能得到像平均曲率一样的奇异性信息。
We study minimization problems on Hardy–Sobolev type inequality. We consider the case where singularity is in interior of bounded domain Ω ⊂ R^ N Ω⊂ RN. The attainability of best constants for Hardy–Sobolev type inequalities with boundary singularities have been studied so far, for example Ghoussoub and Kang (Ann Inst Henri Poincare Anal Non Lineaire 21 (6): 767–793, 2004), Ghoussoub and Robert (IMRP 21867: 1–85, 2006), Ghoussoub and Robert (Trans Am Math Soc 361 (9): 4843–4870, 2009) etc.... According to their results, the mean curvature of ∂ Ω∂ Ω at singularity affects the attainability of the best constants. In contrast with boundary singularity case, in interior singularity case it is well known that the best Hardy–Sobolev constant μ _s (Ω):=\left {∫ _ Ω| ∇ u|^ 2 dx\Bigg| u ∈ H_0^ 1 (Ω),\int _ Ω| u|^ 2^*(s)| x|^ s dx= 1\right\} μ s (Ω):=∫ Ω|∇ u| 2 dx| u∈ H 0 1 (Ω),∫ Ω| u| 2∗(s)| x| sdx= 1 is never achieved for all bounded domain Ω Ω. We can see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding H^ 1 (Ω) ↪ L^ 2^*(s)(Ω,| x|^-s dx) H 1 (Ω)↪ L 2∗(s)(Ω,| x|-sdx) for bounded domain Ω Ω with 0 ∈ Ω 0∈ Ω. In this problem, scaling invariance doesn’t hold and we can not obtain information of singularity like mean curvature.