From forms to semigroups

TL;DR

This paper introduces a new approach to form methods for solving evolutionary problems on Hilbert spaces, simplifying the theory by removing the need for closability and demonstrating efficiency for Dirichlet-to-Neumann operators and degenerate equations.

Contribution

It presents a reformulation of form methods that eliminates the need for the concept of closability, enhancing applicability to certain operators and equations.

Findings

Simplified approach to form methods without closability

Effective application to Dirichlet-to-Neumann operators

Direct proof of holomorphic semigroup generation

Abstract

Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on [AE2]. The main point in our approach is that the notion of closability is not needed anymore. The new setting is particularly efficient for the Dirichlet-to-Neumann operator and degenerate equations. Besides this we give several other examples. This presentation starts by an introduction to holomorphic semigroups. Instead of the contour argument found in the literature, we give a more direct argument based on the Hille--Yosida theorem.