On the minimum problem for non-quasiconvex vectorial functionals

Abstract

We consider functionals of the form F(u)=∫Ωf(x,u(x),Du(x))dx,u∈u0+W01,r(Ω,Rm),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {F}}(u) = \\int _{\\Omega } f(x, u(x), D u(x))\\,dx, \\quad u\\in u_0 + W_0^{1,r}(\\Omega ,{\\mathbb {R}}^m), \\end{aligned}$$\\end{document}where the integrand f:Omega times {mathbb {R}}^mtimes {mathbb {M}}^{mtimes n} rightarrow {mathbb {R}} is assumed to be non-quasiconvex in the last variable and u_0 in W^{1,r}(Omega ,{mathbb {R}}^m) is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope {overline{f}} of f and of the relaxed functional F¯(u)=∫Ωf¯(x,u(x),Du(x))dx,u∈u0+W01,r(Ω,Rm),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\overline{{\\mathcal {F}}}(u) = \\int _{\\Omega } {\\overline{f}}(x, u(x), D u(x))\\,dx, \\quad u\\in u_0 + W_0^{1,r}(\\Omega ,{\\mathbb {R}}^m), \\end{aligned}$$\\end{document}imposing standard differentiability and growth properties on {overline{f}}. In addition we assume a suitable structural condition on {overline{f}} and a special regularity on the minimizers of overline{{mathcal {F}}}, showing that under such assumptions {mathcal {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). This last argument does not require the special regularity assumption mentioned above but the usual notion of classical differentiability (almost everywhere).