On optimal L p regularity in evolution equations

Alessandra Lunardi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 1, page 25-34
  • ISSN: 1120-6330

Abstract

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Using interpolation techniques we prove an optimal regularity theorem for the convolution u ⁢ t = ∫ 0 t T ⁢ t - s ⁢ f ⁢ s ⁢ d ⁢ s , 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 ⁢ R n , 1 < p < ∞ , in which case it yields new optimal regularity results in fractional Sobolev spaces.

How to cite

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Lunardi, Alessandra. "On optimal \( L^{p} \) regularity in evolution equations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.1 (1999): 25-34. <http://eudml.org/doc/252383>.

@article{Lunardi1999,
abstract = {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.},
author = {Lunardi, Alessandra},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Abstract evolution equations; Optimal regularity; Interpolation; abstract evolution equations; interpolation; optimal regularity; convolution; strongly continuous semigroup; Ornstein-Uhlenbeck semigroup; fractional Sobolev spaces},
language = {eng},
month = {3},
number = {1},
pages = {25-34},
publisher = {Accademia Nazionale dei Lincei},
title = {On optimal \( L^\{p\} \) regularity in evolution equations},
url = {http://eudml.org/doc/252383},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Lunardi, Alessandra
TI - On optimal \( L^{p} \) regularity in evolution equations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/3//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 1
SP - 25
EP - 34
AB - 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.
LA - eng
KW - Abstract evolution equations; Optimal regularity; Interpolation; abstract evolution equations; interpolation; optimal regularity; convolution; strongly continuous semigroup; Ornstein-Uhlenbeck semigroup; fractional Sobolev spaces
UR - http://eudml.org/doc/252383
ER -

References

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