On lower semicontinuity in the calculus of variations

Leoni, Giovanni

Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 2, pp. 345-364.

Résumé

Vengono studiate proprietà di semicontinuità per integrali multipli $$u\in W^{k, 1} (\Omega; \mathbb{R}^{d})\mapsto \int_{\Omega} f(x, u(x), \ldots \nabla^{k}u(x)) \, dx $$ quando $f$ soddisfa a condizioni di semicontinuità nelle variabili $(x, u, \ldots, \nabla^{k-1}u(x) )$ e può non essere soggetta a ipotesi di coercitività, e le successioni ammissibili in $W^{k, 1} (\Omega; \mathbb{R}^{d})$ convergono fortemente in $W^{k-1, 1} (\Omega; \mathbb{R}^{d})$.

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