Partial Regularity in Problems Motivated by Nonlinear Elasticity

Regularity is proven almost everywhere for minimisers of problems motivated by nonlinear elasticity. Model problems treated include \[ ∫_Ω {| {Du} |^2 + | { Du} |^2 } ,\] where u:Ω ( ⊂ R² ) → R² and \[ ∫_Ω {| {Du} |^2 + | {Du} |^s + | {AdDu} |^s + | { Du} |^2 } ,\] where u:Ω - (R³ ) → R³ 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 → 0.