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In this note, we address the following question: Why certain nonassociativealgebra structures emerge in the regularity theory of elliptic type PDEs andalso in constructing nonclassical and singular solutions? The aim of the paperis twofold. Firstly, to give a survey of diverse examples on nonregularsolutions to elliptic PDEs with emphasis on recent results on nonclassicalsolutions to fully nonlinear equations. Secondly, to define an appropriatealgebraic formalism which makes the analytic part of the construction ofnonclassical solutions more transparent.

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