On a new class of elastic deformations not allowing for cavitation

Abstract Let Ω ⊂ ℝ n be open and bounded and assume that u :Ω → ℝ n satisfies u ∈ W 1, p (Ω, ℝ n ), adj D u ∈ L q (Ω; ℝ n × n ) with p ≧ n − 1, q ≧ n n − 1 . We show that for g ∈ C 1 (ℝ n ; ℝ n ) with bounded gradient, one has the identity ∂ ∂ x j { ( g i ∘ u ) ( adj D u ) i j } = ( div g ) ∘ u det D u in the sense of distributions. As an application, we obtain existence results in nonlinear elasticity under weakened coercivity conditions. We also use the above identity to generalize Sverak’s ( cf . [Sv88]) regularity and invertibility results, replacing his hypothesis q ≧ p p − 1 by q ≧ n n − 1 . Finally if q = n n − 1 and if det D u ≧ 0 a.e., we show that det D u ln (2 + det D u ) is locally integrable.