Strong approximation of fractional Sobolev maps
(2014) Journal of Fixed Point Theory and Applications — Vol. 15, n° 1, p. 133-153 (2014)
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- Abstract
- Brezis and Mironescu have announced several years ago that for a compact manifold N^n contained in the Euclidean space R^nu and for real numbers 0 < s < 1 and p greater than or equal to 1 the class of smooth maps on the cube with values into N^n is dense with respect to the strong topology in the fractional Sobolev space W^{s, p}(Q^m; N^n) when the homotopy group of N^n of order sp is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that N^n is sp connected, we give a shorter and different proof of their result. Our proof for sp greater than or equal to 1 is based on the existence of a retraction of R^nu onto N^n except for a small subset in the complement of N^n and on the Gagliardo-Nirenberg interpolation inequality for maps in W^{1, q} cap L^infty. In contrast, the case sp < 1 relies on the density of step functions on cubes in W^{s, p}.
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Bousquet, P., Ponce, A., & Van Schaftingen, J. (2014). Strong approximation of fractional Sobolev maps. Journal of Fixed Point Theory and Applications, 15(1), 133-153. https://doi.org/10.1007/s11784-014-0172-5 (Original work published 2014)