Lunardi: On optimal \( L^{p} \) regularity in evolution equations

Lunardi, Alessandra:
On optimal \( L^{p} \) regularity in evolution equations (Regolarità ottimale \( L^{p} \) in equazioni di evoluzione)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 10 (1999), fasc. n.1, p. 25-34, (English)
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Usando tecniche di interpolazione si dimostra un teorema di regolarità ottimale per la convoluzione \( u(t) = \int_{0}^{t} T(t-s) f(s) ds \), dove \( T(t) \) è un semigruppo fortemente continuo in uno spazio di Banach qualunque. Nel caso dei problemi parabolici astratti – cioè quando \( T(t) \) è un semigruppo analitico – esso permette di ritrovare in modo unificato risultati di regolarità già noti. Il teorema può essere applicato anche nel caso di alcuni semigruppi non analitici, come ad esempio la realizzazione del semigruppo di Ornstein-Uhlenbeck in \( L^{p} (\mathbb{R}^{n}) \), \( 1 < p < \infty \), per il quale dà nuovi risultati di regolarità ottimale in spazi di Sobolev frazionari.

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Referenze Bibliografiche