Category:Definitions/Sobolev Spaces

This category contains definitions related to Sobolev Spaces.
Related results can be found in Category:Sobolev Spaces.

Let $\openint a b$ be an open real interval.

Let $\map {L^2} {a, b}$ be the Lebesgue space.

Let $f \in \map {L^2} {a, b}$.

Suppose that in the sense of distributions we have that:

$f, \ldots, f^{\paren n} \in \map {L^2} {a, b}$

where $n \in \N$.

Then the Sobolev space is defined and denoted as:

This article, or a section of it, needs explaining. In particular: Presumably the "Sobolev space on $\map {L^2} {a, b}$"? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

$\map {H^n} {a, b} := \set {f \in \map {L^2} {a, b} : f', \ldots, f^{\paren n} \in \map {L^2} {a, b} }$

and equipped with the Sobolev norm.