Towards nonlinearity. The p-regularity theory. Applications and developments
E. Bednarczuk, O. Brezhneva, K. Leśniewski, A. Prusińska, A. Tret'yakov
TL;DR
The paper addresses nonlinear equations and equality‑constrained optimization that are nonregular (degenerate) by developing the theory of $p$‑regularity, which replaces a nononto derivative with a surjective $p$‑factor operator built from higher‑order derivatives. It develops generalized Lyusternik–Graves/implicit function results, a $p$‑factor Newton method, and $p$‑regular optimality conditions, enabling robust solutions in degenerate settings. These tools yield both existence results and practical algorithms for degenerate problems arising in physics, engineering, and applied analysis, with potential applicability to nonlinear PDEs, control, geometry, and variational problems. The framework thus extends classical regularity theory to ill‑posed contexts, providing stable convergence guarantees and new interpolation and variational techniques for a broad class of nonlinear problems.
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 Poincaré'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.