On the unique extension problem for functionals of the calculus of variations

Carbone, Luciano ; De Arcangelis, Riccardo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 12 (2001) no. 2, pp. 85-106

Résumé

By drawing inspiration from the treatment of the non parametric area problem, an abstract functional is considered, defined for every open set in a given class of open subsets of $\mathbb{R}^{n}$ and every function in $C^{\infty} (\mathbb{R}^{n})$, and verifying suitable assumptions of measure theoretic type, of invariance, convexity, and lower semicontinuity. The problem is discussed of the possibility of extending it, and of the uniqueness of such extension, to a functional verifying analogous properties, but defined in wider families of open sets and less smooth functions. A suitable extension is constructed, and minimal sufficient conditions for its uniqueness are proposed. The results are applied to some examples in Calculus of Variations.