Lower semicontinuity for quasiconvex integrals of higher order

Abstract

We consider a functional of the type F(u) = Ω F (x, u, . . . , D k u) dx, where Ω is an open bounded set of R n and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of F with respect to the weak topology of W k,p (Ω; R m ) under p-growth conditions for the integrand F .

k-quasi-convexity reduces to quasi-convexity

“…We give here a proof of this fact (see Proposition 2.7), that was already known in the cases k = 1 (see [17]) and k = 2 (see [14]). Thanks to Theorem 1.2, the study of lower semicontinuity of (1.1) reduces to a first order problem.…”

“…In [18], the author proved that k-quasiconvexity is a necessary and sufficient condition for sequential lower semicontinuity of (1.1) with respect to weak convergence in the Sobolev space W k,p (Ω), under appropriate p-growth and continuity conditions on the integrand f . This result has been later extended to the case where f is a Carathéodory integrand by Fusco (see [13]) and by Guidorzi and Poggiolini (see [14]), for p = 1 and p > 1 respectively.…”

A-Quasiconvexity: Relaxation and Homogenization

“…the work of Acerbi and Fusco [1], Dacorogna [13], Marcellini and Sbordone [28] and the references contained therein). When s > 1 lower semicontinuity results related to Theorem 1.3 are due to Meyers [29], Fusco [23] and Guidorzi and Poggiolini [25], while we are not aware of any integral representation formula for the relaxed energy, when the integrand depends on the full set of variables, that is f = f (x, u, . .…”

Partial regularity of minimizers of higher order integrals with (p,q)-growth

“…For a higher order version of this result we refer to Meyers [39] and Guidorzi and Poggiolini [28]. Assuming additionally the following coercivity condition…”