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

Abstract

We investigate a known problem whether a Sobolev homeomorphism between domains in $$\mathbb {R}^n$$ can change sign of the Jacobian. The only case that remains open is when $$f\in W^{1,[n/2]}$$ , $$n\ge 4$$ . We prove that if $$n\ge 4$$ , and a sense-preserving homeomorphism f satisfies $$f\in W^{1,[n/2]}$$ , $$f^{-1}\in W^{1,n-[n/2]-1}$$ and either f is Holder continuous on almost all spheres of dimension [n / 2], or $$f^{-1}$$ is Holder continuous on almost all spheres of dimensions $$n-[n/2]-1$$ , then the Jacobian of f is non-negative, $$J_f\ge 0$$ , 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.