Rigidity of Euclidean product structure: breakdown for low Sobolev exponents.

The purpose of this paper is twofold. First, we show that the results in a companion paper on product rigidity for maps $ f: \Omega_1 \times \Omega_2 \subset \mathbb R^n \times \mathbb R^n \to \mathbb R^n \times \mathbb R^n $ in the Sobolev space $ W^{1, p} $ are sharp with respect to $ p $. Specifically, we show that for all $ n \ge 2 $ and all $ p < 2 $ there exist maps $ f \in W^{1, p} $ such that the weak differential $ Df $ is invertible almost everywhere and preserves or reverses the product structure almost everywhere, but $ f $ is not of the form $ f(x, y) = (f_1(x), f_2(y)) $ or = $ f(x, y) = (f_2(y), f_1(x)) $. Secondly, we develop a general toolbox to study $ W^{1, p} $ solutions of differential inclusions $ Du \in K $ for unbounded sets $ K $. As an illustration we give short proofs of results by Astala et al. on optimal $ W^{1, p} $ regularity for solutions of elliptic equations with measurable coefficients and results by Colombo and Tione on irregular solutions of the $ p $-Laplace equation. [ABSTRACT FROM AUTHOR]