Journal of Convex Analysis 33 (2026), No. 3&4, 953--982
Copyright Heldermann Verlag 2026
Some Remarks on Quasiconvexity
Paolo Marcellini
Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Firenze, Italy
paolo.marcellini@unifi.it
This article is dedicated to Hedy Attouch, colleague and friend, with whom I shared many studies on Gamma convergence and other types of variational convergence [see the author, Su una convergenza di funzioni convesse, Boll. Unione Mat. Italiana 8 (1973) 137--158], as well described in his book [see H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman, Boston (1984)]. I have a vivid memory of the discussions we had together on these and other topics in calculus of variations.
This manuscript deals with some properties of quasiconvex functions and their applications to problems of the calculus of variations, in particular lower semicontinuity, existence and regularity -- partial and everywhere regularity -- of minimizers of convex and quasiconvex energy integrals under p,q-growth conditions. Motivated by a model energy integral relevant in nonlinear elasticity and by the phenomenon of cavitation, we point out the interest to study lower semicontinuity and relaxation in the context of subcritical growth. Subcritical growth appears to be a context for a real challenge on quasiconvexity.
Keywords: Quasiconvex functions, subcritical growth, cavitation, p,q-growth conditions, calculus of variations, elliptic systems.
MSC: 49J45; 35J20, 49N60.
[ Fulltext-pdf (218 KB)] for subscribers only.