Elliptic operators and maximal regularity on periodic little-H\"older spaces

TL;DR

This paper establishes maximal regularity for inhomogeneous parabolic equations with elliptic operators in periodic little-Hölder spaces, under minimal coefficient regularity, and extends results to operator-valued coefficients.

Contribution

It proves continuous maximal regularity for elliptic operators in periodic little-Hölder spaces with minimal assumptions and extends to vector-valued operator coefficients.

Findings

Maximal regularity holds in periodic little-Hölder spaces.

Invertibility and resolvent bounds are established for parameter-dependent elliptic equations.

Results extend to elliptic operators with operator-valued coefficients.

Abstract

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H\"older spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.