Critical-exponent Jacobian sign problem in Sobolev homeomorphisms

Determine whether the sign of the Jacobian of Sobolev homeomorphisms is preserved in the limiting case p = floor(n/2) for dimensions n >= 4, thereby completing the characterization of when the Jacobian can change sign.

Background

The introduction summarizes prior results showing that, under specified integrability conditions, the Jacobian of Sobolev homeomorphisms cannot change sign, while in dimensions n >= 4 sign changes are possible when 1 <= p < floor(n/2). The borderline exponent p = floor(n/2) is identified as unresolved in the cited literature. This question concerns the general Sobolev setting and is not resolved by the BV results established in the paper.