EXISTENCE OF MINIMIZERS FOR NON-QUASICONVEX FUNCTIONALS BY STRICT MONOTONICITY

We consider functionals of the form F(u) = integral(Omega )f (x, u(x), Du(x)) dx, u is an element of u(0)+W-0(1,r)(Omega,R-m), where the integrand f = f (x, p, xi) : Omega x R-m x M-mxn -> R is assumed to be non-quasiconvex in the last variable and u0 is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope f of f and of the relaxed functional F(u) = integral(Omega )f(x, u(x), Du(x)) dx, u is an element of u(0 )+ W-0( 1,r )(Omega,R-m),imposing standard differentiability and growth properties on f. Then we assume the quasiaffinity of f on the set in which f > f and the strict monotonicity of the map R is not an element of p(i) (sic) f (x, p, xi), where p(i) is a single scalar component of the vector function variable p, showing that any minimizer of F minimizes F too.