Inégalité de Kato

En analyse fonctionnelle, l'inégalité de Kato est une inégalité de distribution pour l'opérateur de Laplace ou dans le cas général certains opérateurs elliptiques. Elle a été prouvée en 1972 par le mathématicien japonais Tosio Kato^[1].

On traite ici le cas particulier de l'opérateur de Laplace.

Inégalité de Kato

Soit $\Omega \subset \mathbb {R} ^{d}$ un ensemble ouvert et borné et $f\in L_{\operatorname {loc} }^{1}(\Omega )$ , de sorte que $\Delta f\in L_{\operatorname {loc} }^{1}(\Omega )$ . On a alors^[2]^,^[3]

\Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad

\;{\mathcal {D}}'(\Omega )

où^[4]

\operatorname {sgn} {\overline {f}}={\begin{cases}{\frac {\overline {f(x)}}{|f(x)|}}&{\text{si }}f\neq 0\\0&{\text{si }}f=0.\end{cases}}

Explications

$L_{\operatorname {loc} }^{1}$ est espace des fonctions localement intégrable.

Références

↑ (en) Tosio. Kato, « Schrödinger operators with singular potentials », Israel J. Math., vol. 13,‎ 1972, p. 135–148 (DOI 10.1007/BF02760233)
↑ W. Arendt et A.F.M. ter Elst, Kato’s Inequality, vol. 146, Springer, coll. « Analysis and Operator Theory. Springer Optimization and Its Applications », 2019 (DOI 10.1007/978-3-030-12661-2_3)
↑ Haı̈m Brezis et Augusto Ponce, Kato's inequality when $\Delta u$ is a measure, vol. 338, coll. « Comptes Rendus de l'Academie des Sciences - Series I: Mathematics », 2004 (DOI 10.1016/j.crma.2003.12.032), p. 599-604
↑ (en) Toshio Horiuchi, « Some remarks on Kato’s inequality », Journal of Inequalities and Applications,‎ 2001 (DOI 10.1155/S1025583401000030)

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