Friedrichs's inequality

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In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp 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. Friedrichs's inequality generalizes the Poincaré–Wirtinger inequality, which deals with the case k = 1.

Statement of the inequality

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Let   be a bounded subset of Euclidean space   with diameter  . Suppose that   lies in the Sobolev space  , i.e.,   and the trace of   on the boundary   is zero. Then  

In the above

  •   denotes the Lp norm;
  • α = (α1, ..., αn) is a multi-index with norm |α| = α1 + ... + αn;
  • Dαu is the mixed partial derivative  

See also

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References

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  • Rektorys, Karel (2001) [1977]. "The Friedrichs Inequality. The Poincaré inequality". Variational Methods in Mathematics, Science and Engineering (2nd ed.). Dordrecht: Reidel. pp. 188–198. ISBN 1-4020-0297-1.