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Minimum Value for c in the Sobolev Inequality $\| {\phi ^3 } \|\leqq c\| {\nabla \phi } \|^3 $

1971; Society for Industrial and Applied Mathematics; Volume: 21; Issue: 1 Linguagem: Inglês

10.1137/0121004

ISSN

1095-712X

Autores

Gerald Rosen,

Tópico(s)

Stability and Controllability of Differential Equations

Resumo

Previous article Next article Minimum Value for c in the Sobolev Inequality $\| {\phi ^3 } \|\leqq c\| {\nabla \phi } \|^3 $Gerald RosenGerald Rosenhttps://doi.org/10.1137/0121004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIt is shown that the minimum value for c is $4/3 \sqrt 3 \pi ^2 \cong .0780$ for $\phi $ of function class $C^0 $ piecewise $C^2 $ in real Euclidean 3-space.[1] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115–162 MR0109940 0088.07601 Google Scholar[2] Hiroshi Fujita and , Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269–315 10.1007/BF00276188 MR0166499 0126.42301 CrossrefISIGoogle Scholar[3] Hiroshi Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem, J. Fac. Sci. Univ. Tokyo Sect. I, 9 (1961), 59–102 (1961) MR0132307 0111.38502 Google Scholar[4] R. Courant and , D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953xv+561 MR0065391 0053.02805 Google Scholar[5] E. Kamke, Differentialgleichungen Lösungsmethoden and Lösungen, Akademische Verlags, Leipzig, 1956 Google Scholar[6] M. A. Krasnoselskii˘, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by LeoF. Boron, P. Noordhoff Ltd. 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