Discrete Fourier multipliers and cylindrical boundary value problems

TL;DR

This paper investigates operator-valued boundary value problems with periodic conditions in multi-dimensional domains, establishing conditions for unique solutions and applying these results to vector-valued parabolic problems with maximal regularity.

Contribution

It introduces discrete Fourier multiplier techniques to analyze boundary value problems and extends the results to cylindrical domains with applications to parabolic equations.

Findings

Equivalent conditions for unique solvability of boundary value problems

Maximal L^q-regularity for vector-valued parabolic problems

Application of discrete Fourier multipliers in boundary value analysis

Abstract

We consider operator-valued boundary value problems in $(0,2\pi)^n$ with periodic or, more generally, $\nu$-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains $(0,2\pi)^n\times V$ with $\nu$-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal $L^q$-regularity for such problems.