Regulaarsusteooria

Regulaarsusteooria, also known as regularity theory, is a field of mathematics that studies the properties of solutions to certain types of differential equations and geometric partial differential equations. It focuses on understanding when these solutions exhibit smooth, predictable, or otherwise "regular" behavior. This regularity can manifest in various ways, such as differentiability, the absence of singularities, or specific geometric shapes.

The theory originated from attempts to understand the behavior of minimal surfaces and solutions to the Poisson equation. Over time, it has expanded to encompass a wide range of problems, including those arising in fluid dynamics, elasticity, and image processing. Key questions in regularity theory involve determining how the regularity of the input data (e.g., the coefficients of an equation or the boundary conditions) affects the regularity of the solution. It also investigates how certain geometric constraints or physical principles can impose regularity on otherwise potentially irregular solutions.

A central theme in regularity theory is the use of various analytical tools, such as Sobolev spaces, variational methods, and geometric measure theory, to prove the existence and smoothness of solutions. The findings of regularity theory are crucial for both theoretical understanding and the development of numerical methods for solving these complex equations. Understanding the regularity of solutions allows for more reliable approximations and predictions in scientific and engineering applications.