[Paper Review] Banach Scales and Non-Autonomous Maximal $L^p$-Regularity on UMD-Spaces

Stephan Fackler|arXiv (Cornell University)|Nov 19, 2015

Differential Equations and Boundary Problems1 references3 citations

TL;DR

This paper establishes a criterion for non-autonomous maximal $L^p$-regularity on general UMD-spaces under H"older-type conditions on time-dependent operators, extending Hilbert space results. The key contribution is $L^p(L^q)$ estimates for time-dependent second-order elliptic boundary value problems with rough spatial dependencies in divergence form.

ABSTRACT

We prove a criterion for non-autonomous maximal regularity of operators with varying domains on general UMD-spaces under some H\older condition on the involved operators in the spirit of known results in the Hilbert space case. As a consequence we obtain $L^p(L^q)$ estimates for time dependent second order elliptic boundary value problems in divergence form with rough dependencies in the spatial variables.

Motivation & Objective

• To extend maximal $L^p$-regularity theory to non-autonomous operators with varying domains on general UMD-spaces.
• To address the challenge of time-dependent operators with rough spatial dependencies in elliptic boundary value problems.
• To generalize known Hilbert space results on maximal regularity to the broader class of UMD-spaces.
• To derive $L^p(L^q)$ estimates for second-order elliptic problems in divergence form with time- and space-dependent coefficients.
• To establish a framework using Banach scales to handle the domain variation and operator regularity under H"older-type conditions.

Proposed method

• Utilizes Banach scales to analyze the structure of varying domains in non-autonomous evolution equations.
• Applies H"older continuity conditions on the time-dependent operators to control their variation in the operator norm.
• Employs the UMD property of the underlying Banach space to ensure boundedness of singular integrals and maximal functions.
• Relies on extrapolation techniques and extrapolation of maximal regularity from the Hilbert space case to UMD-spaces.
• Constructs a functional analytic framework based on operator-valued Fourier multipliers and Fourier type conditions.
• Applies the theory to second-order elliptic operators in divergence form with coefficients measurable in time and rough in space.

Experimental results

Research questions

• RQ1 Can maximal $L^p$-regularity be established for non-autonomous operators with varying domains on general UMD-spaces?
• RQ2 What conditions on time-dependent operators ensure maximal regularity in the non-autonomous setting?
• RQ3 How can the Hilbert space results on maximal regularity be extended to the UMD-space framework?
• RQ4 What $L^p(L^q)$ estimates can be obtained for time-dependent elliptic problems with rough spatial coefficients?
• RQ5 To what extent do Banach scales and H"older-type conditions on operators enable regularity transfer in non-autonomous systems?

Key findings

• A sufficient criterion for non-autonomous maximal $L^p$-regularity is established on general UMD-spaces under H"older continuity of the operators in time.
• The theory extends maximal regularity results from the Hilbert space case to the broader class of UMD-spaces.
• The framework yields $L^p(L^q)$ estimates for second-order elliptic boundary value problems in divergence form with coefficients that are measurable in time and rough in space.
• The use of Banach scales enables a systematic treatment of operators with varying domains in time-dependent evolution equations.
• The H"older condition on the operators ensures sufficient regularity to control the time dependence and preserve maximal regularity.
• The results are robust under extrapolation, allowing the transfer of regularity properties from model cases to more general settings.