On the Relaxation and the Lavrentieff Phenomenon for Variational Integrals with Pointwise Measurable Gradient Constraints ∗ Riccardo De Arcangelis Universit` a di Napoli “Federico II” Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” via Cintia, Complesso Monte S. Angelo, 80126 Napoli E-mail: dearcang@unina.it Sara Monsurr` o Universit` a di Napoli “Federico II” Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” via Cintia, Complesso Monte S. Angelo, 80126 Napoli E-mail: monssara@matna2.dma.unina.it Elvira Zappale Universit` a di Salerno Dipartimento di Ingegneria dell’Informazione e Matematica Applicata via Ponte don Melillo, 84084 Fisciano (Sa) E-mail: zappale@diima.unisa.it 1 Introduction Let U be a set, and G : U → [0, +∞]. At a general level, the relaxation process for G consists in the introduction on U of a suitable topology, say τ , and in the determination of the sequential τ - lower semicontinuous envelope of G, say G, defined as the greatest sequentially τ -lower semicontinuous functional on U less than or equal to G. This is done in order to approach the minimization problem of G on U since, if suitable coerciveness conditions on G hold, it turns out that G has minima on U , and min U G = inf U G. Let now Ω be a smooth bounded open subset of R N , and let us denote by L(Ω) the σ-algebra of the Lebesgue measurable subsets of Ω. Let g :Ω × R N → [0, +∞[ (1.1) be a Carath´ eodory function, i.e. such that g(·,z) is L(Ω)-measurable for every z ∈ R N and g(x, ·) is continuous for a.e. x ∈ Ω. In addition, let us take p ∈ [1, +∞], and assume that, if p ∈ [1, +∞[ g(x, z) ≤ Λ(a(x)+ |z| p ) for a.e. x ∈ Ω and every z ∈ R N , (1.2) ∗ This work is part of the European Research Training Network “Homogenization and Multiple Scales” (HMS 2000), under contract HPRN-2000-00109 1