ON A NEW CLASS OF ELASTIC DEFORMATIONS NOT ALLOWING FOR CAVITATION

Let OMEGA subset-of R(n) be open and bounded and assume that u: OMEGA --> R(n) satisfies u is-an-element-of W1, W(p) (OMEGA, R(n)), adj Du i s-an-element-of L(q) (OMEGA; R(n x n)) with p greater-than-or-equal-to n - 1, q greater-than-or-equal-to n/n-1. We show that for g is-an-ele ment-of C1(R(n); R(n)) with bounded gradient, one has the identity par tial derivative/partial derivative x(j){(g(i).u)(adj Du)i(j)} = (div g ).u det Du in the sense of distributions. As an application, we obtain existence results in nonlinear elasticity under weakened coercivity c onditions. We also use the above identity to generalize Sverak's (cf. [Sv88]) regularity and invertibility results, replacing his hypothesis q greater-than-or-equal-to p/p-1 by q greater-than-or-equal-to n/n-1. Finally if q = n/n-1 and if det Du greater-than-or-equal-to 0 a.e., w e show that det Du ln(2 + detDu) is locally integrable.