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,
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}\left(1+{\Bigl (}\log {\bigl (}1+{\frac {\|u\|_{H^{2}(\Omega )}}{\|u\|_{H^{1}(\Omega )}}}{\bigr )}{\Bigr )}^{1/2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5acde23b8c5cefa9bdd338077f576d6c3c2d84)
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:
![{\displaystyle {\begin{aligned}\displaystyle \|{\widehat {v}}\|_{L^{1}(\mathbb {R} ^{2})}&=\int _{|\xi |<R}|{\widehat {v}}(\xi )|{\rm {d}}\xi +\int _{|\xi |>R}|{\widehat {v}}(\xi )|{\rm {d}}\xi \\&=\int _{|\xi |<R}(1+|\xi |)|{\widehat {v}}(\xi )|{\frac {1}{1+|\xi |}}{\rm {d}}\xi +\int _{|\xi |>R}(1+|\xi |^{2})|{\widehat {v}}(\xi )|{\frac {1}{1+|\xi |^{2}}}{\rm {d}}\xi \\&\leq C\left(\int _{|\xi |<R}{\frac {1}{(1+|\xi |)^{2}}}{\rm {d}}\xi \right)^{\frac {1}{2}}+C\|u\|_{H^{2}(\Omega )}\left(\int _{|\xi |>R}{\frac {1}{(1+|\xi |^{2})^{2}}}{\rm {d}}\xi \right)^{\frac {1}{2}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e38c76051fee9c6d5757e8dbefd6c661caa1b54f)
owing to the preceding inequalities and to the Cauchy-Schwarz inequality. This yields
![{\displaystyle \|{\widehat {v}}\|_{L^{1}(\mathbb {R} ^{2})}\leq C(\log(1+R))^{\frac {1}{2}}+C{\frac {\|u\|_{H^{2}(\Omega )}}{1+R}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120d8257715be1bea656ad8605c8204a17b6b54b)
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
![{\displaystyle \int _{\mathbb {R} ^{2}}{\bigl (}(\partial _{11}^{2}v)^{2}+2(\partial _{12}^{2}v)^{2}+(\partial _{22}^{2}v)^{2}{\bigr )}=\int _{\mathbb {R} ^{2}}{\bigl (}\partial _{11}^{2}v+\partial _{22}^{2}v{\bigr )}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707403084dcfa990766ce076d4f7e0a448c8f466)
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,
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}\left(1+{\Bigl (}\log {\bigl (}1+{\frac {\|\Delta u\|_{L^{2}(\Omega )}}{\|u\|_{H^{1}(\Omega )}}}{\bigr )}{\Bigr )}^{1/2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c72421236cbf13e8c5de631af34d0d9007a4c177)
The previous inequality is close to the way that the Brezis-Gallouet inequality is cited in.[2]
See also
- Ladyzhenskaya inequality
- Agmon's inequality
References
- ^ 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
- ^ Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. (2001). Navier–Stokes Equations and Turbulence. Cambridge: Cambridge University Press. ISBN 0-521-36032-3.