Summary: A Regularity Theorem
for Minimizersof QuasieonvexIntegrals
EMILIO ACERBI & NICOLA Fusco
Communicated by E. GIUST!
We prove C 1'~ partial regularity for minimizers of functionals with quasi-
convex integrand f(x, u, Du) depending on vector-valued functions u. The inte-
grand is required to be twice continuously differentiable in Du, and no assump-
tion on the growth of the derivatives off is made: a polynomial growth is required
only on f itself.
Consider the functional I(u) = ff(Du(x)) dx, where s is an open subset
and f: R "N--->~.
The regularity of minimizers of I has been widely investigated (see  and
its extensive bibliography), but until recently the function f was required to be
convex, which rules out many interesting physical examples (see ) and is far
from quasiconvexity (this condition is necessary and sufficient for the semiconti-
nuity of I on appropriate Sobolev spaces, see , and so it is a fundamental