# Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs

## TL;DR

This paper explores the spectral properties of nonassociative algebras and their role in the regularity theory of nonlinear elliptic PDEs, providing a survey and an algebraic framework for understanding nonclassical solutions.

## Contribution

It introduces an algebraic formalism that clarifies the emergence of nonassociative algebra structures in elliptic PDE regularity theory and surveys recent developments in nonregular solutions.

## Key findings

- Identification of algebraic structures in PDE regularity theory
- Survey of recent nonclassical solutions to fully nonlinear equations
- Development of an algebraic formalism for PDE solution analysis

## Abstract

In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Secondly, to define an appropriate algebraic formalism which makes the analytic part of the construction of nonclassical solutions more transparent.