On minimization problems which approximate Hardy Lp inequality

✍ Scribed by Arkady Poliakovsky

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 238 KB
• Volume: 54
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00130-5

✦ Synopsis

Let be a smooth bounded domain in R N with 0 ∈ and let p ∈ (1; ∞){N }. By a classical inequality of Hardy we have |∇v| p ¿ c * p; N |v| p =|x| p , for all 0 = v ∈ W 1;p 0 ( {0}), with c * p; N = |(N -p)=p| p being the best constant in this inequality. More generally, for Á ∈ C( ) such that Á ¿ 0; Á = 0 and Á(0) = 0 we have, for certain values of , that |∇v| p -Á|v| p =|x| p ¿ c * p; N |v| p =|x| p , for all 0 = v ∈ W 1;p 0 ( \ {0}).

In particular, it follows that there is no minimizer for this inequality. We consider then a family of approximating problems, namely inf 0 =v∈W 1;p 0 ( {0}) |∇v| p -Á|v| p =|x| p |v| p-=|x| p

for ¿ 0, and study the asymptotic behavior, as → 0, of the positive minimizers {u } which are normalized by u p = 1. We prove the convergence u → u * in 1¡q¡p W 1;q 0 ( \ {0}), where u * is the unique positive solution (up to a multiplicative factor) of the equationpu = (u p-1 =|x| p )(c * p; N + Á(x)) in {0}, with u = 0 on 9 .