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Wirtinger-Sobolev Isoperimetric Constants

Wirtinger-Sobolev isoperimetric constants are the best constants gamma such that

[int_Omega|f|^qdx]^(1/q)<=gamma[int_Omegasum_(i=1)^N|(partialf)/(partialx_i)|^pdx]^(1/p),
(1)

where f is a real-valued smooth function on a region Omega satisfying some boundary conditions.

For real-valued absolutely continuous functions on [0,1] satisfying f(0)=f(1)=0, with 1<p<infty, 1<=q<infty, and f^' in L^p(0,1), the sharp constant in

(int_0^1|f(x)|^qdx)^(1/q)<=C_(p,q)(int_0^1|f^'(x)|^pdx)^(1/p)
(2)

is

C_(p,q)=q/2(1+r/q)^(1/p)(1+q/r)^(-1/q)(Gamma(1/q+1/r))/(Gamma(1/q)Gamma(1/r)),
(3)

where r=p/(p-1) and Gamma is the gamma function (Dacorogna et al. 1992, pp. 31 and 40). For p=q=2, this gives C_(2,2)=1/pi, the endpoint form of Wirtinger's inequality.

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References

Dacorogna, B.; Gangbo, W.; and Subía, N. "Sur une généralisation de l'inégalité de Wirtinger." Ann. Inst. H. Poincaré C Anal. Non Linéaire 9, 29-50, 1992. https://doi.org/10.1016/S0294-1449(16)30249-9.Finch, S. R. "Sobolev Isoperimetric Constants." §3.6 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 219-225, 2003.

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Wirtinger-Sobolev Isoperimetric Constants

Cite this as:

Weisstein, Eric W. "Wirtinger-Sobolev Isoperimetric Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Wirtinger-SobolevIsoperimetricConstants.html

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