Friedrichs' inequality について

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Friedrichs' inequality ：ウィキペディア英語版

Friedrichs' inequality
In mathematics, Friedrichs' inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the ''L^p'' norm of a function using ''L^p'' bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent.
==Statement of the inequality==

Let Ω be a bounded subset of Euclidean space R^''n'' with diameter ''d''. Suppose that ''u'' : Ω → R lies in the Sobolev space

W_^ (\Omega)

(i.e. ''u'' lies in ''W''^''k'',''p''(Ω) and the trace of ''u'' is zero). Then
:

\| u \|_ \leq d^ \left( \sum_ \| \mathrm^ u \|_^ \right)^.

In the above
*

\| \cdot \|_

denotes the ''L^p'' norm;
* ''α'' = (''α''₁, ..., ''α''_''n'') is a multi-index with norm |''α''| = ''α''₁ + ... + ''α''_''n'';
* D^α''u'' is the mixed partial derivative
::

\mathrm^ u = \frac} \cdots \partial_} }.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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