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Poincaré 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 Poincaré inequality says that there exist constants C_1 and C_2 such that

 int_Omegag^2(x)dx<=C_1int_Omega|del g(x)|^2dx+C_2[int_Omegag(x)dx]^2

for all functions g in the Sobolev space H^1(Omega) consisting of all functions in L^2(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 piecewise on Omega.

In some literature, the above-stated Poincaré inequality is sometimes referred to as the mean Poincaré inequality with the unqualified phrase "Poincaré inequality" reserved for the so-called (and closely-related) Friedrichs inequality. Inequalities which are qualitatively similar to those of Friedrichs and Poincaré are sometimes referred to collectively as Poincaré-Friedrichs inequalities.


See also

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

This entry contributed by Christopher Stover

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References

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

Cite this as:

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

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