## Advances in Differential Equations

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

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

#### 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