Definition:Sobolev Norm

Definition

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

Let $\map {H^n} {a, b}$ be the Sobolev space.

Let $f \in \map {H^n} {a, b}$.

Let $\norm {\,\cdot\,}_p$ be the $2$-seminorm.

Then the Sobolev norm, denoted by $\norm {\,\cdot\,}_{k,p}$, is defined by

$\ds \norm {f}_{k,p} := \paren {\sum_{i \mathop = 0}^k \norm {f^{\paren i} }_p^p}^{\frac 1 p}$

Source of Name

This entry was named for Sergei Lvovich Sobolev.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense