泛函的约束变分问题

The Confined Variation Problem of The Functional

本文讨论Ｊλ＝ｉｎｆΩ（｜Ｄｕ｜ｐ＋ａ（ｘ）｜ｕ｜ｐ）｛Ω｜ｕ｜ｐ＊＝λ，ｕ∈ｗ１，ｐ（Ω）｝的可达性，其中Ω是Ｒｎ中具有ｃ１边界的有界区域，ｎ≥２，１＜ｐ＜ｎ，ｐ＊＝ｎｐ／（ｎ－ｐ），ａ（ｘ）∈ｃ（Ω），且ａ（ｘ）≥０，ａ（ｘ）０，λ＞０．

n this paper, we discuss the problem of the confined variation J λ= inf {Ω(|Du| p+a(x)|u| p)|Ω|u| p * =λ, u∈w 1,p (Ω)} where Ω is a bounded domain in R n with c′ boundary. n≥2, 1<p<n, p *=np/(n-p), a(x)∈C(), and a(x)≥0, a(x)0, λ>0.