Lifting in Sobolev Spaces
نویسندگان
چکیده
for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this paper we study the question of lifting in the framework of the Sobolev spaces W s,p with 0 < s < ∞ and 1 < p < ∞. The motivation comes from problems of the Ginzburg-Landau type where one considers questions such as Min ∫ |∇u| in the class of functions u : Ω → S (see e.g. F. Bethuel, H. Brezis and F. Hélein [1]).
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