𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Analyticity and Discrete Maximal Regularity on Lp-Spaces

✍ Scribed by Sönke Blunck

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
189 KB
Volume
183
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

✦ Synopsis

In the first part of this paper, we give the following interpolation result on the analyticity (i.e. the property &(T&I ) T n & CÂn for all n # N) of an operator T on L p : If T is powerbounded on L p and L q as well as analytic on L p , then T is powerbounded and analytic on L r for all r strictly between p and q. This is a discrete analogue of the well-known corresponding result for analytic semigroups (e tA ).

As recently shown by the author, the analyticity of T is a necessary condition for the maximal regularity of the discrete time evolution equation u n+1 &Tu n = f n for all n # Z + , u 0 =0. In the second part of this paper we establish the following two sufficient conditions for its maximal regularity: T is a subpositive analytic contraction, or T is an integral operator satisfying certain Poisson bounds. These results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u$(t)&Au(t)= f (t) for all t # R + , u(0)=0, due to Lamberton, Weis, Coulhon and Duong and Hieber and Pru ss. For the Poisson bound result of Coulhon and Duong and Hieber and Pru ss we give a slight improvement and a short proof.

📜 SIMILAR VOLUMES

Maximal Lp regularity for parabolic diff
Maximal Lp regularity for parabolic difference equations
✍ Matthias Geissert 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 174 KB

## Abstract In this paper, we consider a family of finite difference operators {__Ah__ }~__h__ >0~ on discrete __L__ ~__q__~ ‐spaces __L__ ~__q__~ (ℝ^__N__^ ~__h__~ ). We show that the solution __u__ ~__h__~ to __u__ ′~__h__~ (__t__) – __A__ ~__h__~ __u__ ~h~(__t__) = __f__ ~__h__~ (__t__), __t__

Non Integral Regularizing Operators on L
Non Integral Regularizing Operators on Lp- Spaces
✍ Jürgen Voigt 📂 Article 📅 1999 🏛 John Wiley and Sons 🌐 English ⚖ 157 KB

## Abstract We present bounded positivity preserving operators from __L__~__p__~(ℝ) to __L__~__q__~ (__ℝ__), for 1 < __p__ < ∞, 1/p‐1/q < 1/2, which are not integral operators.

On the Lp–Lq maximal regularity for the
On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain
✍ Yuka Naito 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 261 KB

## Abstract This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: __u__|=__D__~ν~__u__|=θ|=0 and initial condition: (__u, u__~__t__~, θ)|~__t__=0~=(__u__~0~, __v__~0~, θ~0~). Here, Ω is a bounded domain in ℝ^__n__^(__n__≧2). We assume tha

On the Lp–Lq maximal regularity for Stok
On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain
✍ Rieko Shimada 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 328 KB

## Abstract We obtain the __L__~__p__~–__L__~__q__~ maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ℝ^__n__^ (__n__⩾2). The Robin condition consists of two conditions: __v__ ⋅ __u__=0 and α__u__+β(__T__(__u__, __p__)__v__ – 〈__T__(__u__, __p__)__v__,