Hölder space

The Hölder space ( after Otto Hölder ) is in mathematics, a Banach space of functions, which plays a role in the theory of partial differential equations. There Hölder spaces are a natural choice to operate existence theory can.

Definition

Be. The Hölder space is the set of all functions, is responsible for the following standard finite:

Referred to here

The supremum norm and

Half standard. For one writes.

The Hölder space is also the space of - times continuously differentiable, bounded functions from to, the -th partial derivatives are Holder continuous on a constant and also limited. In the special case, we speak mostly of Lipschitz continuity.

Set of Kellogg

Be and be a bounded domain with boundary and a strictly elliptic operator in with coefficients in, ie

Which are located in the matrix and the ellipticity

Met with an independent constants. Next, the function is not positive and well. Then has the equation

A unique classical solution.

Since the above equation has no classical solution, if continuity is required by only, is to check the continuity modulus of relevance for the existence of theory in the theory of partial differential equations. Hölder spaces are a class of functions that can be operated within which classical theory of existence.