A new obstruction to the extension problem for Sobolev maps between manifolds

作者：Bethuel Fabrice^*

来源：Journal of Fixed Point Theory and Applications, 2014, 15(1): 155-183.

DOI：10.1007/s11784-014-0185-0

• Sobolev maps
• extension problem

The main result of the present paper, combined with earlier results of Hardt and Lin [Comm. Pure Appl. Math. 40 (1987), 555-588], settles the extension problem for , where and are compact Riemannian manifolds, having nonempty smooth boundary and assuming, moreover, that is simply connected. The main question we study is the following: Given a map in the trace space W (1-1/p,p) (), does it possess an extension in W (1,p) ()? We show that the answer is negative in the case , where the number is related to the topology of . We also address the case is not simply connected, providing various results and rising some open questions. In particular, we stress in that case the relationship between the extension problem and the lifting problem to the universal covering manifold.

• 出版日期2014-3