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Functionals with p x growth and regularity

Emilio AcerbiGiuseppe Mingione — 2000

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the integral functional f x , D u d x under non standard growth assumptions of p , q -type: namely, we assume that z p x f x , z L 1 + z p x , a relevant model case being the functional D u p x d x . Under sharp assumptions on the continuous function p x > 1 we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.

A variational problem for couples of functions and multifunctions with interaction between leaves

Emilio AcerbiGianluca CrippaDomenico Mucci — 2012

ESAIM: Control, Optimisation and Calculus of Variations

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy...

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