Regularity for a class of variational integrals motivated by non-linear elasticity

Abstract

23 Fuchs, M., Regularity for a class of variational integrals motivated by non-linear elasticity, Asymptotic Analysis 9 (1994) 23-38. We amsider (local) minimizers u E H1,P(SI,lRN ) of variational integrals F(u) := In f(Du)dx with integrand f coming asymptotically close to IDulP for large values of Du and prove everywhere regularity theorems without imposing any differentiability or convexity conditions on f. Further sections contain applications to certain relaxed problems. o. Introduction Motivated by the theory of hyperelastic materials in non-linear elasticity (see [1, 2, 6, 18]) one is led to the study of polyconvex stored energy densities1 f(Du) = w (Du, adjz Du, ... , adjmin(n,N) DU) (adjs Du:= matrix of the s x s minors of Du) on suitable classes of deformations u: lR,n :> {} -+ lR,N where in physical applications {} is a domain in lR,3 representing the undeformed state of the body and N equals 3. Polyconvexity of f means that w is a convex function of the different minors which in addition has to satisfy suitable growth conditions, e.g., min(n,N) w( ... ) ~ IDulP + L asladjs Dul a • + b, (0.1) s=1 p > 1, as ~ 1, with real constants as and b. From the physical point of view one also imposes the hypothesis (n = N) f(Du) -+ 00 as det Du -+ 0 (0.2) 1 For simplicity we consider homogeneous materials. Correspondence to: M. Fuchs, Fachbereich Mathematik, Universitat des Saarlandes, D-66041 Saarbrticken, Germany. 0921-7134/94/$06.00 © 1994 Elsevier Science B.V. All rights reserved 24 M. Fuchs / Regularity for a class of variational integrals which means that one needs an infinite amount of energy to shrink a finite volume to zero. Under the above conditions equilibrium states of the body can be found by looking at the variational problem F(u, 12):= In I(Du) dx --+ Min in C (0.3) where C denotes a subclass of a suitable Sobolev space taking care of the growth conditions (0.1), (0.2) and where also various boundary conditions as well as non-linear constraints as det Du > 0 are incorporated. Then, under rather general hypothesis, Ball [1, 2] (compare also the references in [8]) proved lower semi-continuity theorems thus establishing the existence of solutions to (0.3) which are continuous if one works in H 1,P(Q, IRN) for p > n = dim Q. In case 1 O. The most general results concern local minimizers of functionals F(u, 12) = In I(Du) dx where I has to satisfy i) I E c2(lRnN), ii) I grows of order I Du IP, e.g., min.0!,N) I/(Du)1 ~ IDulP + ~ csladjsDul a • + Co with exponents as n 1. In our paper we consider the regularity problem for stored energy densities I which become asymptotically close to IDul P for large values of Du, but we impose neither differentiability (f E CO is sufficient) nor any convexity conditions on the integrand I, more precisely we require that lim IAI-P I(A) IAI-+oo exists in (0,00) for some exponent p > 1. For example we can discuss densities of the form (n = N = 3) I(Du) = IDul P + a 'IDul q + b ·ladj2 Dul a + c·1 detDul i3 + d(o + I detDun-t, p > 1, 0 0 which include the integrands studied in [16, 17] as special cases but nevertheless we benefited from the arguments of Malek-Madani and Smith. It should be noted that there are materials whose stored energy density fails to be quasi-convex (see [8] and the references quoted therein) so that one is led to the study of the associated relaxed problem. As we shall show in Section 4 our results apply to generalized minimizers (Le. weak limit points of minimizing sequences). We shall prove that (generalized) minimizers are Holder continuous on the whole domain 12 which is similar to a result of Chi pot and Evans [5] who discussed integrands of class C2 becoming convex at infinity. M. Fuchs / Regularity for a class of variational integrals 25 1. Notations, examples and statement of the results Let n c IRn, n ~ 2, denote an open bounded region and fix an integer N. If A E IRnN we write A = (A~h~a~n l~i~N in particular if u: n -+ IRN we abbreviate Du = (Dauih~a~n,. l~i~N On the space IR nN we consider the Euclidean norm Suppose that p E (1, (0) is a real number and let I: IRnN -+ IR denote a continuous function satisfying the limit condition lim IAI-P I(A) = 1. IAI-+oo (1.1) Then the variational integral (G a subregion of n with compact closure)