Regularitätsklassen

Regularitätsklassen are a concept in mathematical analysis, particularly in the study of partial differential equations. They are used to classify the regularity or smoothness of solutions. Instead of a binary distinction between "smooth" and "not smooth," regularity classes provide a nuanced framework. Solutions are assigned to a specific class based on the differentiability of their derivatives. A common notation for these classes is $C^k$, where $k$ represents the order of differentiability. A function in $C^k$ is continuously differentiable up to its $k$-th derivative. Higher values of $k$ indicate greater smoothness. The class $C^\infty$ denotes infinitely differentiable functions, also known as smooth functions. There are also more refined regularity classes, such as Sobolev spaces, which incorporate differentiability in a weaker sense, allowing for solutions that may not be classically differentiable everywhere but still possess useful properties. The precise definition and properties of a given regularity class are crucial for determining the existence, uniqueness, and behavior of solutions to differential equations. Understanding these classes helps mathematicians analyze how information propagates through a system described by the equation and predict the qualitative features of its solutions.