Weighted interpolation inequalities: a perturbation approach
Weighted interpolation inequalities: a perturbation approach
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加权插值不等式：扰动方法
DOI：
10.1007/s00208-016-1480-4
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发表时间：
2015
影响因子：
1.4
通讯作者：
B. Nazaret
中科院分区：
文献类型：
--
作者：
J. Dolbeault;Matteo Muratori;B. Nazaret
在标准对称化方法失效的条件下，我们研究了一族具有幂函数权的Caffarelli-Kohn-Nirenberg不等式族的最优函数。我们证明了最优函数的存在性，研究了它们的性质，并证明了当权的幂足够小时，它们是径向的。径向对称到平移对于重量为零的极限情况是正确的，这种情况对应于著名的Gagliardo-Nirenberg不等式子族。我们的方法是基于浓度紧致性分析和使用谱间隙不等的微扰方法。因此，我们证明了最优函数是显式的，并且在微扰区由Barenblatt型轮廓给出。
We study optimal functions in a family of Caffarelli–Kohn–Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo–Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.