Some isoperimetric inequalities on RN with respect to weights |x|α

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:451

Article

Alvin, A.¹  Brock, F.²  Chiacchio, F.¹  Mercaldo, A.¹  Posteraro, M. R.¹ 

[1] Univ Naples Federico II, Dipartimento Matemat & Applicaz R Caccioppoli, Complesso Monte S Angelo,Via Cintia, I-80126 Naples, Italy

[2] Univ Rostock, Dept Math, Ulmenstr 69, D-18057 Rostock, Germany

关键词: Isoperimetric inequality;    Weighted rearrangement;    Norm inequality;    Elliptic boundary value problem;    Eigenvalue problem;   

DOI  :  10.1016/j.jmaa.2017.01.085

来源: Elsevier

【 摘 要 】

We solve a class of isoperimetric problems on R-N with respect to weights that are powers of the distance to the origin. For instance we show that, if k is an element of [0,1], then among all smooth sets Omega in R-N with fixed Lebesgue measure, fan integral partial derivative Omega vertical bar x vertical bar k N-1(dx) achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya Szego principle. In turn, we establish radiality of optimizers in some Caffarelli Kohn Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free