Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$

W. Arendt, A. F. M. ter Elst, M. Sauter

Abstract

Nittka gave an efficient criterion on a form defined on $L_2(Ω)$ which implies that the associated semigroup is $L_p$-invariant for some given $p \in (1,\infty)$. We extend this criterion to the Hilbert space valued~$L_2(Ω,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_p$-contractive for all $p \in [p_-,p_+]$ for some $1 < p_- < 2 < p_+ < \infty$.

Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$

Abstract

Nittka gave an efficient criterion on a form defined on which implies that the associated semigroup is -invariant for some given . We extend this criterion to the Hilbert space valued~. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is -contractive for all for some .
Paper Structure (5 sections, 15 theorems, 44 equations)

This paper contains 5 sections, 15 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Nittka's criterion for $L_p$-contractivity
  3. Application to Hilbert space valued parabolic problems
  4. The derivative of a truncation
  5. Strict convexity of $L_p(\Omega,H)$

Key Result

Theorem 1.1

Let $(\Omega,{\cal B},\mu)$ be a $\sigma$-finite measure space. Let $H$ be a Hilbert space. Fix $p \in (1,\infty)$. Define Let $P$ be the orthogonal projection of $L_2(\Omega,H)$ onto $C$. Let ${\cal V}$ be a Hilbert space which is continuously and densely embedded in $L_2(\Omega,H)$. Let $\gothic{a} \colon {\cal V} \times {\cal V} \to \mathds{C}$ be a continuous elliptic sesquilinear form, let $

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['tpacc101']}.
  • Theorem 3.1
  • proof
  • ...and 21 more