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

Abstract

In this paper, 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.

A note on 2-generated symmetric axial algebras of Monster type

“…Assume first that i + 1 ≡ 3 0. Substituting the expressions ( 6) and (7) in Equation (9), with h = 1 and k = i, and using Equation (14) we get…”

“…Axial algebras of Monster type have been introduced in [4] by Hall, Rehren and Shpectorov, in order to generalise some subalgebras of the Griess algebra (defined Majorana algebras by Ivanov [5]) and create a new tool for better understanding, and possibly unifying, the classification of finite simple groups. They have recently appeared also in other branches of mathematics (see [8,9]). In [6,7] Rehren started a systematic study of primitive 2-generated axial algebras of Monster type, constructing several classes of new algebras.…”

Fractional-Order Adaptive P -Laplace Equation-Based Art Image Edge Detection

“…In the literature [5], a nonlinear anisotropic combined diffusion model was designed; this model is mainly by choosing different diffusion methods when facing different regions of the image, homogeneous diffusion for smooth regions, and mean curvature diffusion for edge regions, and the effectiveness of this model was experimentally demonstrated. The literature [6] combined the concept of fractional order to improve a model of anisotropic diffusion, which also led to the research on the application of fractional order in the field of image edge detection. The literature [7] proposed an improved image edge detection model with anisotropic diffusion by different diffusion coefficient functions.…”

V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras

“…For example, coming back to isoparametric Cartan cubic algebras, the Peirce spectrum contains essentially two elements (distinct from the trivial eigenvalue 1) with simple fusion rules. This makes it easy to get a complete classification (see § 3 in [57]). A similar situation occurs for classical (formal real) Jordan algebras or axial algebras of Jordan type [11], [12].…”

“…The interested reader is referred to the recent monograph [31] for unexpected connections of nonassociative algebras to regularity theory of fully nonlinear PDEs. See also the recent papers [52], [55], [56], [18], [58], [57] for further results and basic concepts considered below.…”