Towards nonlinearity. The p-regularity theory. Applications and developments

TL;DR

This paper reviews recent advances in p-regularity theory, a framework for analyzing nonlinear equations with singular operators, and demonstrates its applications in solving degenerate problems across various scientific fields.

Contribution

It introduces new numerical methods based on p-regularity for solving degenerate nonlinear equations and optimization problems, expanding the toolkit for handling singularities in complex systems.

Findings

Development of the first numerical approaches for degenerate problems

Application of p-regularity to diverse fields like materials and geophysics

Enhanced understanding of tangent cones in singular nonlinear equations

Abstract

We present recent advances in the analysis of nonlinear equations with singular operators and nonlinear optimization problems with constraints given by singular mappings. The results are obtained within the framework of $p$-regularity theory, which has developed successfully over the last forty years. We illustrate the theory with its applications to degenerate problems in various areas of mathematics. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in a singular case. The structure of p-factor operators is used to propose optimality conditions and construct numerical methods for solving degenerate nonlinear equations and optimization problems. The methods presented in the paper can be considered as the first numerical approaches targeting solutions of degenerate problems, such as the Van der Pol differential equation, boundary-value problems with a small parameter, partial differential equations where Poincar\'e's method of small parameter fails, nonlinear degenerate dynamical systems, and others. There are various practical applications for the theory of p-regularity, including structural engineering, composite materials, and material design. For instance, the theory can be applied to analyze the behavior of materials with irregular or complex properties. By considering higher-order derivatives, it becomes possible to model and predict the response of materials to external forces, such as stress or temperature variations. In geophysics, the $p$-regularity theory can be utilized to analyze and interpret complex data obtained from seismic surveys, gravity measurements, or electromagnetic surveys. The theory also finds applications in the analysis of nonlinear differential equations arising in control systems, geometric and topological analysis, biomechanics, and many other fields.