Partial Regularity in Problems Motivated by Nonlinear Elasticity
Nicola Fusco, John Hutchinson · SIAM Journal on Mathematical Analysis · 1991
Regularity is proven almost everywhere for minimisers of problems motivated by nonlinear elasticity. Model problems treated include \[ \int_\Omega {| {Du} |^2 + | {\det Du} |^2 } ,\] where $ u:\Omega ( \subset \mathbb{R}^2 ) \to \mathbb{R}^2 $ and \[ \int_\Omega {| {Du} |^2 + | {Du} |^s + | {AdDu} |^s + | {\det Du} |^2 } ,\] where $ u:\Omega - (\mathbb{R}^3 ) \to \mathbb{R}^3 $ with $s > 2$. In particular, continuity of minimisers is not assumed a priori. “Degenerate” convexity of the integrand in the higher minors M of $Du$ is also allowed, in the sense that second derivatives in M may approach zero as $M \to 0$.