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On the Caffarelli‐Kohn‐Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions
Author(s) -
Catrina Florin,
Wang ZhiQiang
Publication year - 2001
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/1097-0312(200102)54:2<229::aid-cpa4>3.0.co;2-i
Subject(s) - nirenberg and matthaei experiment , mathematics , embedding , constant (computer programming) , symmetry (geometry) , inequality , pure mathematics , mathematical physics , mathematical analysis , combinatorics , geometry , computer science , programming language , artificial intelligence
Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]$$\left(\int_{\Re^N} |x|^{-bp}|u|^{p} \,dx \right)^{2/p} \leq C_{a,b} \int_{\Re^N} |x|^{-2a}|\nabla u|^2 \,dx$$ where, for N ≥ 3, −∞ < a < ( N − 2)/2, a ≤ b ≤ a + 1, and p = 2 N /( N − 2 + 2( b − a )). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given. © 2001 John Wiley & Sons, Inc.