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

Lunardi, Alessandra

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 1, pp. 25-34.

Abstract (VO)

Using interpolation techniques we prove an optimal regularity theorem for the convolution \( u(t) = \int_{0}^{t} T(t-s) f(s) ds \), where \( T(t) \) is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when \( T(t) \) is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in \( L^{p} (\mathbb{R}^{n}) \), \( 1 p \infty \), in which case it yields new optimal regularity results in fractional Sobolev spaces.

Riassunto

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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