On the minimum problem for non-quasiconvex vectorial functionals

Abstract

AbstractWe consider functionals of the form $$\begin{aligned} {\mathcal {F}}(u) = \int _{\Omega } f(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega ,{\mathbb {R}}^m), \end{aligned}$$ F ( u ) = ∫ Ω f ( x , u ( x ) , D u ( x ) ) d x , u ∈ u 0 + W 0 1 , r ( Ω , R m ) , where the integrand $$f:\Omega \times {\mathbb {R}}^m\times {\mathbb {M}}^{m\times n} \rightarrow {\mathbb {R}}$$ f : Ω × R m × M m × n → R is assumed to be non-quasiconvex in the last variable and $$u_0 \in W^{1,r}(\Omega ,{\mathbb {R}}^m)$$ u 0 ∈ W 1 , r ( Ω , R m ) is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope $${\overline{f}}$$ f ¯ of f and of the relaxed functional $$\begin{aligned} \overline{{\mathcal {F}}}(u) = \int _{\Omega } {\overline{f}}(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega ,{\mathbb {R}}^m), \end{aligned}$$ F ¯ ( u ) = ∫ Ω f ¯ ( x , u ( x ) , D u ( x ) ) d x , u ∈ u 0 + W 0 1 , r ( Ω , R m ) , imposing standard differentiability and growth properties on $${\overline{f}}$$ f ¯ . In addition we assume a suitable structural condition on $${\overline{f}}$$ f ¯ and a special regularity on the minimizers of $$\overline{{\mathcal {F}}}$$ F ¯ , showing that under such assumptions $${\mathcal {F}}$$ F attains its infimum. Futhermore, we study the minimum problem for a class of functionals with separate dependence on the gradients of competing maps by the use of integro-extremality method, proving an existence result inspired by analogous ones obtained in the scalar case ($$m=1$$ m = 1 ). This last argument does not require the special regularity assumption mentioned above but the usual notion of classical differentiability (almost everywhere).