Quasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers

Abstract

We establish C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{C}^{\\infty }$$\\end{document}-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u;Ω]:=∫ΩF(∇u)dx,u:Ω→RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathscr {F}[u;\\Omega ]:=\\int _{\\Omega }F(\ abla u)\ extrm{d}x,\\qquad u:\\Omega \\rightarrow \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}subject to a q-growth condition |F(z)|≦c(1+|z|q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|F(z)|\\leqq c(1+|z|^{q})$$\\end{document}, z∈RN×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\mathbb {R}^{N\ imes n}$$\\end{document}, and natural p-mean coercivity conditions on F∈C∞(RN×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F\\in \ extrm{C}^{\\infty }(\\mathbb {R}^{N\ imes n})$$\\end{document} for the basically optimal exponent range 1≦p≦q<min{npn-1,p+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\leqq p\\leqq q<\\min \\{\\frac{np}{n-1},p+1\\}$$\\end{document}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).