Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 1, page 29-33
  • ISSN: 1120-6330

Abstract

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We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space C - 1 , α ⁢ ( Ω ) consinsting of all derivatives of hölder-continuous functions in Ω where Ω is a domain in ℝ n not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space C - 1 , α ⁢ ( Ω ) . We prove also that the spaces C - 1 , α ⁢ ( Ω ) can be considered as extrapolation spaces relative to suitable non-variational operators.

How to cite

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Vespri, Vincenzo. "Analytic semigroups generated on a functional extrapolation space by variational elliptic equations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.1 (1988): 29-33. <http://eudml.org/doc/287477>.

@article{Vespri1988,
abstract = {We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^\{-1,\alpha\} (\Omega)$ consinsting of all derivatives of hölder-continuous functions in $\Omega$ where $\Omega$ is a domain in $\mathbb\{R\}^\{n\}$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^\{-1,\alpha\} (\Omega)$. We prove also that the spaces $C^\{-1,\alpha\} (\Omega)$ can be considered as extrapolation spaces relative to suitable non-variational operators.},
author = {Vespri, Vincenzo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Analytic semigroups; Elliptic variational; Extrapolation spaces; unbounded domain; infinitesimal generator; analytic semigroup; Hölder- continuous functions; interpolation spaces},
language = {eng},
month = {3},
number = {1},
pages = {29-33},
publisher = {Accademia Nazionale dei Lincei},
title = {Analytic semigroups generated on a functional extrapolation space by variational elliptic equations},
url = {http://eudml.org/doc/287477},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Vespri, Vincenzo
TI - Analytic semigroups generated on a functional extrapolation space by variational elliptic equations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/3//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 1
SP - 29
EP - 33
AB - We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^{-1,\alpha} (\Omega)$ consinsting of all derivatives of hölder-continuous functions in $\Omega$ where $\Omega$ is a domain in $\mathbb{R}^{n}$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^{-1,\alpha} (\Omega)$. We prove also that the spaces $C^{-1,\alpha} (\Omega)$ can be considered as extrapolation spaces relative to suitable non-variational operators.
LA - eng
KW - Analytic semigroups; Elliptic variational; Extrapolation spaces; unbounded domain; infinitesimal generator; analytic semigroup; Hölder- continuous functions; interpolation spaces
UR - http://eudml.org/doc/287477
ER -

References

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  1. CANNARSA, P. - VESPRI, V. - Generation of Analytic Semigroups by Elliptic Operators with Unbounded Coefficients, «Siam J. Math. Anal.», to appear. Zbl0623.47039MR883572DOI10.1137/0518063
  2. DA PRATO, G. and GRISVARD, P. (1984) - Maximal Regularity for Evolution Equations by Interpolation and Extrapolation, «J. Funct. Anal.», 58, 107-124. Zbl0593.47041MR757990DOI10.1016/0022-1236(84)90034-X
  3. LIONS, J.L. and PEETRE, J. (1964) - Sur une classe d'espace d'interpolation. «Publ. I.H.E.S. Paris», 5-68. Zbl0148.11403MR165343
  4. TRIEBEL, H. (1978) - Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam. Zbl0387.46032MR503903
  5. VESPRI, V. (1987) - The Functional Space C - 1 , α ⁢ ( Ω ) and Analytic Semigroups. Preprint II Università di Roma, June 1987. Zbl0743.47021MR945822

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