# On Time-Periodic Solutions to Parabolic Boundary Value Problems of Agmon-Douglis-Nirenberg Type

## TL;DR

This paper investigates time-periodic solutions to parabolic PDEs of Agmon-Douglis-Nirenberg type, providing explicit formulas and coercive Lp estimates that extend classical elliptic results to the time-periodic setting.

## Contribution

It introduces explicit solution formulas and coercive Lp estimates for time-periodic parabolic problems, extending classical elliptic theory to this new context.

## Key findings

- Explicit formulas for solutions in whole- and half-space cases
- Coercive Lp estimates for time-periodic solutions
- Extension of elliptic theory results to parabolic time-periodic problems

## Abstract

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit formula for the solution and establish coercive Lp estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg for elliptic problems to the time-periodic case.