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

2014/02/19 by Fabrice Béthuel, Bethuel, Fabrice · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1402.4614

Abstract

The main result of the present paper, combined with earlier results of Hardt\nand Lin settles the extension problem for W^1,p( mathcal M, mathcal N),\nwhere mathcal M and mathcal N are compact riemannian manfolds, mathcal\nM having non-empty smooth boundary and assuming moreover that mathcal N is\nsimply connected. The main question which is studied is the following: Given a\nmap in the trace space W^{1-\(1)/(p), p} (\∂ mathcal M, mathcal\nN), does it possess an extension in W^1,p( mathcal M, mathcal N)? We show\nthat the answer is negative in the case mathfrak p_c +1\≤ p<m= rm dim\n , mathcal M, where the number mathfrak p_c is related to the topology\nof mathcal N. We also adress the case mathcal N is not simply connected,\nproviding various results and rising some open questions. In particular, we\nstress in that case the relationship between the extension problem and the\nlifting problem to the universal covering manifold.\n

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