Friedrichs inequality

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An inequality of the form


where is a bounded domain of points in an -dimensional Euclidean space with an -dimensional boundary satisfying a local Lipschitz condition, and the function (a Sobolev space).

The right-hand side of the Friedrichs inequality gives an equivalent norm in . Using another equivalent norm in , one obtains (see [2]) a modification of the Friedrichs inequality of the form


There are generalizations (see [3][5]) of the Friedrichs inequality to weighted spaces (see Weighted space; Imbedding theorems). Suppose that and that the numbers , and are real, with being a natural number and . One says that if the norm

is finite, where

and is distance function from to .

Suppose that is a natural number such that

Then, if , , , for the following inequality holds:

where is the derivative of order with respect to the interior normal to at the points of .

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


The constant is independent of throughout.

The inequality is named after K.O. Friedrichs, who obtained it for , (see [1]).


[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. D.F. KalinichenkoN.V. Miroshin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098