Fonseca, Irene and Fusco, Nicola and Marcellini, Paolo:
Topological degree, Jacobian determinants and relaxation
Bollettino dell'Unione Matematica Italiana Serie 8 8-B (2005), fasc. n.1, p. 187-250, Unione Matematica Italiana (English)
pdf (670 Kb), djvu (836 Kb). | MR2122983 | Zbl 1177.49066
Sunto
Si ottiene una caratterizzazione della variazione totale $TV(u, \Omega)$ del determinante Jacobiano $\det Du$ per alcune classi di applicazioni $u : \Omega \rightarrow \mathbb{R}^{n}$ che non fanno parte della tradizionale classe di Sobolev $W^{1, n}(\Omega; \mathbb{R}^{n})$. In particolare, si forniscono formule esplicite per applicazioni localmente Lipschitziane al di fuori di un punto isolato $x_{0}\in \Omega$. Si stabiliscono anche alcune relazioni fra $TV(u, \Omega)$ e il determinante distribuzionale $\text{Det}\, Du$. Inoltre si fornisce una rappresentazione integrale per l'energia rilassata di certi integrali policonvessi relativi ad applicazioni $u\in W^{1, p}(\Omega; \mathbb{R}^{n})\cap W^{1, \infty}(\Omega\setminus \{x_0\}; \mathbb{R}^{n})$.
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