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

Résumé

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.