Brezis–Gallouët inequality

In mathematical analysis, the Brezis–Gallouët inequality,[1] named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on the second derivatives. It is useful in the study of partial differential equations.

Let be the exterior or the interior of a bounded domain with regular boundary, or itself. Then the Brezis–Gallouët inequality states that there exists a real only depending on such that, for all which is not a.e. equal to 0,

Proof

The regularity hypothesis on is defined such that there exists an extension operator such that:

  • is a bounded operator from to ;
  • is a bounded operator from to ;
  • the restriction to of is equal to for all .

Let be such that . Then, denoting by the function obtained from by Fourier transform, one gets the existence of only depending on such that:

  • ,
  • ,
  • .

For any , one writes:

owing to the preceding inequalities and to the Cauchy-Schwarz inequality. This yields

The inequality is then proven, in the case , by letting . For the general case of non identically null, it suffices to apply this inequality to the function .

Noticing that, for any , there holds

one deduces from the Brezis-Gallouet inequality that there exists only depending on such that, for all which is not a.e. equal to 0,

The previous inequality is close to the way that the Brezis-Gallouet inequality is cited in.[2]

See also

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References

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  1. ^ H. Brezis and T. Gallouet. Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4 (1980), no. 4, 677–681. doi:10.1016/0362-546X(80)90068-1  
  2. ^ Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. (2001). Navier–Stokes Equations and Turbulence. Cambridge: Cambridge University Press. ISBN 0-521-36032-3.