Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

被引:1
作者
Bhakta, Mousomi [1 ]
Ganguly, Debdip [2 ]
Karmakar, Debabrata [3 ]
Mazumdar, Saikat [4 ]
机构
[1] Indian Inst Sci Educ & Res Pune IISER Pune, Dept Math, Dr Homi Bhabha Rd, Pune 411008, India
[2] Indian Inst Technol Delhi, Dept Math, IIT Campus, New Delhi 110016, India
[3] Tata Inst Fundamental Res, Ctr Applicable Math, GKVK PO,Post Bag 6503, Bangalore 560065, India
[4] Indian Inst Technol, Dept Math, Mumbai 400076, India
关键词
SIGN-CHANGING SOLUTIONS; ELLIPTIC-EQUATIONS; ASYMPTOTIC PROFILES; EXTINCTION PROFILE; CLASSIFICATION; SYMMETRY; THEOREMS;
D O I
10.1007/s00526-024-02878-3
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Consider the Poincare-Sobolev inequality on the hyperbolic space: for every n >= 3 and 1<p <= n+2/n-2, there exists a best constant Sn,p,lambda(B-n)>0 such that Sn,p,lambda(Bn)( integral Bn|u|p+1dvBn)2p+1 <=integral Bn(|del Bnu|2-lambda u2)dvBn, holds for all u is an element of C-c(infinity)(B-n), and lambda <=(n-1)(2)/4, where (n-1)(2)/4 is the bottom of the L2-spectrum of -Delta(Bn). It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635-671, 2008) that under appropriate assumptions on n, p and lambda there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S-n,S-p,S-lambda(B-n). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in R-n by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18-24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780-6797, 2018) to the Poincare-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744-789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.
引用
收藏
页数:47
相关论文
共 71 条
[1]  
Akagi G, 2013, SPRINGER INDAM SER, V2, P1, DOI 10.1007/978-88-470-2841-8_1
[3]   Symmetry and stability of asymptotic profiles for fast diffusion equations in annuli [J].
Akagi, Goro ;
Kajikiya, Ryuji .
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 2014, 31 (06) :1155-1173
[4]   Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations [J].
Akagi, Goro ;
Kajikiya, Ryuji .
MANUSCRIPTA MATHEMATICA, 2013, 141 (3-4) :559-587
[5]  
AUBIN T, 1976, J MATH PURE APPL, V55, P269
[6]  
AUBIN T., 1976, J DIFFERENTIAL GEOM, V11, P573
[7]   A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics [J].
Badiale, M ;
Tarantello, G .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2002, 163 (04) :259-293
[8]  
Benguria RD, 2008, MATH RES LETT, V15, P613
[9]   Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models [J].
Berchio, Elvise ;
Ferrero, Alberto ;
Grillo, Gabriele .
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, 2014, 102 (01) :1-35
[10]  
BERRYMAN JG, 1980, ARCH RATION MECH AN, V74, P379, DOI 10.1007/BF00249681