An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

TL;DR

This paper introduces a new definition of weak solutions for abstract Cauchy problems in Hilbert spaces, discussing conditions for their existence and uniqueness.

Contribution

It proposes a novel definition of weak solutions in Hilbert spaces and analyzes their fundamental properties.

Findings

Established existence conditions for weak solutions.

Proved uniqueness under certain assumptions.

Provided a framework for further analysis of differential equations in Hilbert spaces.

Abstract

In this note we propose a definition of weak solution for an abstract Cauchy problem in a Hilbert space, and we discuss existence and uniqueness results.