TOPICS
Search

Friedrichs Inequality


Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d. In functional analysis, the Friedrichs inequality says that there exists a constant C such that

 int_Omegag^2(x)dx<=Cint_Omega|del g(x)|^2dx

for all functions g in the Sobolev space H_0^1(Omega) consisting of those functions in L^2(Omega) having zero trace on the boundary partialOmega of Omega whose generalized derivatives are all also square integrable.

This inequality plays an important role in the study of both function spaces and partial differential equations. As such, a number of generalizations have been established to domains Omega and functions g which are less well-behaved, e.g., to polyhedral domains Omega and to functions g which only have desirable behavior only piecewise on Omega.

In some literature, the Friedrichs inequality is unfortunately referred to as the Poincaré inequality though it should be differentiated from the closely-related (mean) Poincaré inequality. Inequalities which are qualitatively similar to those of Friedrichs and Poincaré are sometimes referred to collectively as Poincaré-Friedrichs inequalities.


See also

Function Space, L-p-Space, Poincaré Inequality, Poincaré-Friedrichs Inequalities, Sobolev Space

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

References

Vohralík, M. "On the Discrete Poincaré-Friedrichs Inequalities for Nonconforming Approximations to the Sobolev Space H^1." Numer. Func. Anal. and Opt. 26, 925-952, 2005.

Cite this as:

Stover, Christopher. "Friedrichs Inequality." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FriedrichsInequality.html

Subject classifications