Advances in Differential Equations

$H^\infty$-Calculus for cylindrical boundary value problems

Tobias Nau and Jürgen Saal

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for $p$ such that the (weak) Dirichlet Laplacian admits an ${\mathcal{H}}^\infty$-calculus on $L^p(\Omega)$, from $(3+\varepsilon)'<p<3+\varepsilon$ to $(4+\varepsilon)'<p<4+\varepsilon$ for three-dimensional bounded or unbounded Lipschitz cylinders $\Omega$. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.

Article information

Source
Adv. Differential Equations Volume 17, Number 7/8 (2012), 767-800.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355702976

Mathematical Reviews number (MathSciNet)
MR2963804

Zentralblatt MATH identifier
1277.47021

Subjects
Primary: 35J40: Boundary value problems for higher-order elliptic equations 35K52: Initial-boundary value problems for higher-order parabolic systems 35K35: Initial-boundary value problems for higher-order parabolic equations

Citation

Nau, Tobias; Saal, Jürgen. $H^\infty$-Calculus for cylindrical boundary value problems. Adv. Differential Equations 17 (2012), no. 7/8, 767--800. https://projecteuclid.org/euclid.ade/1355702976.


Export citation