Four applications of majorization to convexity in the calculus of variations

The resemblance between the Horn–Thompson theorem and a recent theorem by Dacorogna–Marcellini–Tanteri indicates that Schur-convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank one convexity or quasiconvexity. In Theorem 6.6, we give simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant.Majorization is used in order to give a very short proof of a theorem of Thompson and Freede [R.C. Thompson, L.J. Freede, Eigenvalues of sums of Hermitian matrices III, J. Res. Nat. Bur. Standards B 75B (1971) 115–120], Ball [J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, in: R.J. Knops (Ed.), Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. 1, Res. Notes Math., 17, Pitman, 1977, pp. 187–241], or Le Dret [H. Le Dret, Sur les fonctions de matrices convexes et isotropes, CR Acad. Sci. Paris, Série I 310 (1990) 617–620], concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity.Next we prove (Theorem 7.3) a lower semicontinuity result for functionals with the form ∫Ωw(Dϕ(x))dx, with w(F)=h(lnVF). Here F=RFUF=VFRF is the usual polar decomposition of F∈gl(n,R), and lnVF is Hencky’s logarithmic strain.We close this paper with a compact proof of Dacorogna–Marcellini–Tanteri theorem, based only on classical results about majorization. The mentioned resemblance of this theorem with the Horn–Thompson theorem is thus explained.