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 ) a modification of the Friedrichs inequality of the form
There are generalizations (see –) 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 ).
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Friedrichs inequality. D.F. KalinichenkoN.V. Miroshin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Friedrichs_inequality&oldid=15223