Notes on the Banach-Necas-Babuska theorem and Kato's minimum modulus of operators

TL;DR

This note reviews the Banach-Necas-Babuska theorem and Kato's minimum modulus of operators, illustrating their proofs and applications to parabolic equations, emphasizing their role in establishing well-posedness in variational analysis.

Contribution

It provides a clear review of the proofs of the Closed Range and BNB theorems using Kato's ideas, and demonstrates their application to parabolic equations.

Findings

Proofs of the Closed Range and BNB theorems are clarified.

Application of BNB theorem to establish well-posedness of parabolic equations.

The note corrects previous inaccuracies in the presentation of these theorems.

Abstract

This note was prepared for a lecture given at Kyoto University (RIMS Workshop: "The State of the Art in Numerical Analysis: Theory, Methods, and Applications", November 8-10, 2017). That lecture described the variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations based on an earlier paper by the author (arXiv:1710.10543). I also presented the Banach-Necas-Babuska (BNB) Theorem or the Babuska-Lax-Milgram (BLM) Theorem as the key theorem of our analysis. For proof of the BNB theorem, it is useful to introduce the minimum modulus of operators by T. Kato. This note presents a review of the proofs of Closed Range Theorem and BNB Theorem following the idea of Kato. Moreover, I present an application to BNB theorem to parabolic equations. The well-posedness is proved by BNB theorem. This note is not an original research paper. It includes no new results. This is a revised manuscript and several incorrect descriptions in the original version are fixed.