Partial regularity of minimizers of quasiconvex integrals with subquadratic growth

We prove partial regularity for minimizers of quasiconvex integrals of the form dx where the integral F() has subquadratic growth, ie .Research supported by MURST, Gruppo Nazionale 40%.     

Lower semicontinuity of quasiconvex integrals

Summary New lower semicontinuity results for quasiconvex integrals. In particular, under certain structure conditions and growth 0≤F(ξ)≤C(∣≤∣ ^q ) the functional ∫ _Ω F(∇u)dx is proved to be lower semicontinuous onW ^1,q with respect to the weak convergence inW ^1,p, p≥q−1. Research supported by the grant No. 201/93/2171 of Czech Grant Agency (GAČR) and by the grant No. 364 of Charles University (GAUK). This article was processed by the author using the Springer-VerlagTex P Jourlg marco package 1991.     

Remarks on the lower semicontinuity of quasiconvex integrals

We prove lower semicontinuity of quasiconvex integrals with integrands controlled by the minors.This work has been partially supported by the Ministero dell'Università e della Ricerca Scientifica and by the European Research Project GADGET II     

Lower semicontinuity for quasiconvex integrals of higher order

We consider a functional of the type , where is an open bounded set of and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of with respect to the weak topology of under p-growth conditions for the integrand F. Received November 8, 1997     

Lower semicontinuity for quasiconvex integrals of higher order

We consider a functional of the type , where is an open bounded set of and F is a Carathéodory function. By an approximation argument we prove the lower semincontinuity of with respect to the weak topology of under p-growth conditions for the integrand F. Received November 8, 1997     

Boundary regularity for almost minimizers of quasiconvex variational problems

We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.     

Existence and Lipschitz regularity for minima

We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional

on ( ) for a class of integrands that are convex in and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on .

     

Approximation of quasiconvex functions,and lower semicontinuity of multiple integrals

We study semicontinuity of multiple integrals f(x,u,Du) dx, where the vector-valued function u is defined for with values in ^N. The function f(x,s,) is assumed to be Carathéodory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology of the Sobolev space H^1,p(^N). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them beingconvex andindependent of (x,s) for large values of . In the special polyconvex case, for example if n=N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of H^1,p(ⁿ) for small p, in particular for some p smaller than n.     

On the structure of quasiconvex hulls

We define the set K_{q,e ⊂ K} of quasiconvex extreme points for compact sets K ⊂ M^N×n and study its properties. We show that K_q,e is the smallest generator of Q(K)-the quasiconvex hull of K, in the sense that Q(K_q,e) = Q(K), and that for every compact subset W ⊂ Q(K) with Q(W) = Q(K), K_q,e ⊂ W. The set of quasiconvex extreme points relies on K only in the sense that . We also establish that K_e ⊂ K_q,e, where K_e is the set of extreme points of C(K)-the convex hull of K. We give various examples to show that K_q,e is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K), K_q,e = K_e. We apply the results to the two well and three well problems studied in martensitic phase transitions.     

Global regularity for divergence form elliptic equations on quasiconvex domains

In this paper, we introduce a notion of quasiconvex domain, and show that the global W1,p regularity holds on such domains for a wide class of divergence form elliptic equations. The modified Vitali covering lemma, compactness method and the maximal function technique are the main analytical tools.     

Lipschitz regularity for minima without strict convexity of the Lagrangian

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of among the functions in W^1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function are bounded and the boundary datum u₀ satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of on , whenever they exist, are Lipschitz. A relaxation result follows.     

Partial regularity of minimizers of polyconvex variational integrals

We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C ², strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50     

On extremal points of quasiconvex functions

In this paper we examine the linear sectionwise relative minimums of a quasiconvex function and give a sufficient condition for quasiconvex functions to have a strict global minimum on an open convex set.