# Friedrichs inequality

An inequality of the form

$$\tag{1 } \int\limits _ \Omega f ^ { 2 } \ d \Omega \leq C \left \{ \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} \ d \Omega + \int\limits _ \Gamma f ^ { 2 } d \Gamma \right \} ,$$

where $\Omega$ is a bounded domain of points $x = x ( x _ {1} \dots x _ {n} )$ in an $n$- dimensional Euclidean space with an $( n - 1)$- dimensional boundary $\Gamma$ satisfying a local Lipschitz condition, and the function $f \equiv f ( x) \in W _ {2} ^ {1} ( \Omega )$( a Sobolev space).

The right-hand side of the Friedrichs inequality gives an equivalent norm in $W _ {2} ^ {1} ( \Omega )$. Using another equivalent norm in $W _ {2} ^ {1} ( \Omega )$, one obtains (see [2]) a modification of the Friedrichs inequality of the form

$$\tag{2 } \int\limits _ \Omega f ^ { 2 } \ d \Omega \leq C \left \{ \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} d \Omega + \left ( \int\limits _ \Gamma f d \Gamma \right ) ^ {2} \right \} .$$

There are generalizations (see [3][5]) of the Friedrichs inequality to weighted spaces (see Weighted space; Imbedding theorems). Suppose that $\Gamma \subset C ^ {(} l)$ and that the numbers $r$, $p$ and $\alpha$ are real, with $r$ being a natural number and $1 \leq p < \infty$. One says that $f \in W _ {p, \alpha } ^ {r} ( \Omega )$ if the norm

$$\| f \| _ {W _ {p, \alpha } ^ {r} ( \Omega ) } = \ \| f \| _ {L _ {p} ( \Omega ) } + \| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) }$$

is finite, where

$$\| f \| _ {L _ {p} ( \Omega ) } = \ \left ( \int\limits _ \Omega | f | ^ {p} \ d \Omega \right ) ^ {1/p} ,$$

$$\| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) } = \sum _ {| k | = r } \| \rho ^ \alpha f ^ {(} k) \| _ {L _ {p} ( \Omega ) } ,$$

$$f ^ { ( k) } = \frac{\partial ^ {| k | } f }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } ,\ \ | k | = \sum _ {i = 1 } ^ { n } k _ {i} ,$$

and $\rho = \rho ( x)$ is distance function from $x \in \Omega$ to $\Gamma$.

Suppose that $s _ {0}$ is a natural number such that

$$r - \alpha - { \frac{1}{p} } \leq \ s _ {0} < r - \alpha + 1 - { \frac{1}{p} } .$$

Then, if $\Gamma \subset C ^ {( s _ {0} + 1) }$, $- p ^ {-} 1 < \alpha < r - p ^ {-} 1$, $r/2 \leq s _ {0}$, for $f \in W _ {p, \alpha } ^ {r} ( \Omega )$ the following inequality holds:

$$\| f \| _ {L _ {p} ( \Omega ) } \leq \ C \left \{ \sum _ {l + s < r/2 } \left \| \left ( \left . \frac{\partial ^ {s} f }{\partial n ^ {s} } \ \right | _ \Gamma \right ) ^ {(} l) \ \right \| _ {L _ {p} ( \Gamma ) } + \| f \| _ {\omega _ {p, \alpha } ^ {r} ( \Omega ) } \right \} ,$$

where $( \partial ^ {s} f/ \partial n ^ {s} ) \mid _ \Gamma$ is the derivative of order $s$ with respect to the interior normal to $\Gamma$ at the points of $\Gamma$.

One can also obtain an inequality of the type (2), which has in the simplest case the form

$$\| f \| _ {L _ {p} ( \Omega ) } ^ {p} \leq \ C \left ( \| f \| _ {\omega _ {p, \alpha } ^ {1} ( \Omega ) } ^ {p} + \left | \int\limits _ \Gamma u \tau d \Gamma \ \right | ^ {p} \right ) ,$$

where

$$p , \gamma > 1,\ \ - \frac{1}{p} < \alpha < 1 - \frac{1}{p} - \frac{1} \gamma ,$$

$$\tau \in L _ \gamma ( \Gamma ),\ \int\limits _ \Gamma \tau d \Gamma \neq 0.$$

The constant $C$ is independent of $f$ throughout.

The inequality is named after K.O. Friedrichs, who obtained it for $n = 2$, $f \in C ^ {(} 2) ( \overline \Omega \; )$( see [1]).

#### References

 [1] K.O. Friedrichs, "Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz" Math. Ann. , 98 (1927) pp. 566–575 [2] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) [3] S.M. Nikol'skii, P.I. Lizorkin, "On some inequalities for weight-class functions and boundary-value problems with a strong degeneracy at the boundary" Soviet Math. Dokl. , 5 (1964) pp. 1535–1539 Dokl. Akad. Nauk SSSR , 159 : 3 (1964) pp. 512–515 [4] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) [5] D.F. Kalinichenko, "Some properties of functions in the spaces and " Mat. Sb. , 64 : 3 (1964) pp. 436–457 (In Russian) [6] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) [7] L. Nirenberg, "On elliptic partial differential equations" Ann. Scuola Norm. Sup. Pisa Ser. 3 , 13 : 2 (1959) pp. 115–162 [8] L. Sandgren, "A vibration problem" Medd. Lunds Univ. Mat. Sem. , 13 (1955) pp. 1–84
How to Cite This Entry:
Friedrichs inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Friedrichs_inequality&oldid=46991
This article was adapted from an original article by D.F. KalinichenkoN.V. Miroshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article