Lions' representation theorem and applications

TL;DR

This paper extends Lions' Representation Theorem using operator theory, develops a derivation-based framework for well-posedness, and applies it to non-autonomous evolution equations with novel initial and periodic boundary conditions.

Contribution

It introduces a new operator-theoretical derivation framework based on RTL and applies it to solve non-autonomous evolution equations with innovative boundary conditions.

Findings

Established a derivation theory based on RTL for well-posedness.

Proved well-posedness of non-autonomous evolution equations with new boundary conditions.

Extended the applicability of Lions' Representation Theorem to broader contexts.

Abstract

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.