Regularitätstheoreme

Regularitätstheoreme, or regularity theorems, are a class of results in mathematics that establish conditions under which objects possessing certain properties must also possess other, more specific properties, often related to smoothness or regularity. These theorems are fundamental in various branches of mathematics, including analysis, differential equations, and geometry.

In the context of partial differential equations, regularity theorems address the smoothness of solutions. For instance, a solution to a linear partial differential equation might be known to exist, but a regularity theorem could prove that this solution is, in fact, differentiable or even infinitely differentiable, given sufficient smoothness of the coefficients and the domain. This is crucial because many analytical techniques rely on the differentiability of the functions involved.

Another area where regularity theorems are prominent is in geometric measure theory. Here, they might concern the regularity of minimal surfaces or other geometric objects. A theorem could show that a surface satisfying a certain variational principle, like having minimal area, must be smooth (e.g., a $C^2$ manifold), even if the initial conditions or defining equations do not explicitly guarantee this smoothness.

The core idea behind regularity theorems is often to use the properties of the object itself, combined with the rules or equations it satisfies, to deduce further regularity. This can involve methods like bootstrapping, where a low level of regularity is used to prove a higher level, or by analyzing the local behavior of the object to show that it must conform to smooth structures. The precise statements and proofs of these theorems are highly dependent on the specific mathematical context in which they are applied.