Jacobians of W^1,p homeomorphisms, case p=[n/2]

2018/12/31 by Goldstein, Paweł, Hajłasz, Piotr
#26B10 #46E35 (Primary) #74B20 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1812.11888

Abstract

We investigate a known problem whether a Sobolev homeomorphism between domains in ℝⁿ can change sign of the Jacobian. The only case that remains open is when f∈ W^1,[n/2], n≥ 4. We prove that if n≥ 4, and a sense-preserving homeomorphism f satisfies f∈ W^1,[n/2], f^-1∈ W^1,n-[n/2]-1 and either f is Hölder continuous on almost all spheres of dimension [n/2], or f^-1 is Hölder continuous on almost all spheres of dimensions n-[n/2]-1, then the Jacobian of f is non-negative, J_f≥ 0, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.

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