Density in W-s,W-p (Omega; N)

作者：Brezis Haim; Mironescu Petru^*

来源：Journal of Functional Analysis, 2015, 269(7): 2045-2109.

DOI：10.1016/j.jfa.2015.04.005

• Density
• Fractional Sobolev spaces
• Homogeneous extensions
• Manifold-valued maps

Let Omega be a smooth bounded domain in R-n, 0 < s < infinity and 1 <= p < infinity. We prove that C-infinity(<(Omega)over bar>; S-1) is dense in W-s,W-p(Omega; S-1) except when 1 <= sp < 2 and n >= 2. The main ingredient is a new approximation method for Ws,P-maps when s < 1. With 0 < s < 1, 1 <= p < infinity and sp < n, Omega a ball, and N a general compact connected manifold, we prove that C-infinity((Omega) over bar; N) is dense in W-s,W-p(Omega; N) if and only if pi([sp]) (N) = 0. This supplements analogous results obtained by Bethuel when s = 1, and by Bousquet, Ponce and Van Schaftingen when s = 2,3, ... [General domains Omega have been treated by Hang and Lin when s = 1; our approach allows to extend their result to s < 1.] The case where s > 1, s is not an element of N, is still open.

• 出版日期2015-10-1
• 单位rutgers