Parabolic systems with coupled boundary conditions

TL;DR

This paper studies elliptic operators with operator-valued coefficients and coupled boundary conditions, analyzing well-posedness and properties of associated parabolic problems through operator-theoretic methods.

Contribution

It introduces a framework for analyzing parabolic systems with coupled boundary conditions using operator-theoretic techniques, extending existing theory.

Findings

Established well-posedness of the parabolic problems

Reduced qualitative properties to operator projections

Provided systematic analysis of coupled boundary conditions

Abstract

We consider elliptic operators with operator-valued coefficients and discuss the associated parabolic problems. The unknowns are functions with values in a Hilbert space $W$. The system is equipped with a general class of coupled boundary conditions of the form $f_{|\partial\Omega}\in \mathcal Y$ and $\frac{\partial f}{\partial \nu}\in {\mathcal Y}^\perp$, where $\mathcal Y$ is a closed subspace of $L^2(\partial\Omega;W)$. We discuss well-posedness and further qualitative properties, systematically reducing features of the parabolic system to operator-theoretical properties of the orthogonal projection onto $\mathcal Y$.