Multiplicity of Solutions for Linear Partial Differential Equations Using (Generalized) Energy Operators

TL;DR

This paper explores how energy operators can generate multiple solutions for linear PDEs, especially the wave equation, by analyzing energy spaces and their properties, offering a new perspective beyond classical solution methods.

Contribution

It introduces the concept of solution multiplicity for linear PDEs using energy spaces and operators, extending the traditional solution framework.

Findings

Energy operators define additional solutions for PDEs.

Energy spaces can be empty, indicating solution multiplicity.

Application to wave equation and evanescent waves demonstrates the approach.

Abstract

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.