Hostname: page-component-f554764f5-sl7kg Total loading time: 0 Render date: 2025-04-11T00:03:02.880Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimates for solutions of elliptic equations in a limit case

Published online by Cambridge University Press:  17 April 2009

Ester Giarrusso  and
Guido Trombetti
Ester Giarrusso
Affiliation:
Dipartimento di Matematica ed Applicazioni, “R. Caccioppoli”, Universita di Napoli, via Mezzocannone, 8; 80134 Napoli, Italia
Guido Trombetti
Affiliation:
Dipartimento di Matematica ed Applicazioni, “R. Caccioppoli”, Universita di Napoli, via Mezzocannone, 8; 80134 Napoli, Italia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝn. We prove, that if the right hand side of the equation is an element of H−1, n(Ω), then u belongs to the Orlicz space LΦ where Φ(t) = exp(|t|n/(n−1)) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 3 , December 1987 , pp. 425 - 434
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Adams, R.A., Sobolev Spaces, (Academic Press, New York, San Francisco, London, 1975).Google Scholar
[2]Alvino, A., “Un caso limite della diseguaglianza di Sobolev in spazi di Lorentz”, Rend. Accad. Sci. Fis. Mat. Napoli XLIV (1977), 105–112.Google Scholar
[3]Alvino, A., Lions, P.L. and Trombetti, G., “On optimization problems with prescribed rearrangements”, (to appear).Google Scholar
[4]Alvino, A. and Trombetti, G., “Sulle migliori costanti di maggioraziome per una classe di equazioni ellittiche degeneri”, Ricerche Mat., XXVII (1978), 413–428.Google Scholar
[5]Bandle, C., Isoperimetric Inequalities and Applications, (Pitman, London 1980).Google Scholar
[6]De Giorgi, E., “Su una teoria generale della misura (r−1) dimensionale in uno spazio ad r dimensioni”, Ann. Mat. Pura Appl., 36 (1954), 197–213.CrossRefGoogle Scholar
[7]Fleming, W. and Rishel, R., “An integral formula for total gradient variations, Arch Math., 11 (1960).CrossRefGoogle Scholar
[8]Giarrusso, E. and Nunziante, D., “Symmetrization in a class of first-order Hamilton-Jacobi equations”, Nonlinear Anal. 8 (1984), 289–299.CrossRefGoogle Scholar
[9]Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983).Google Scholar
[10]Kawohl, B., Rearrangements and Convexity of Level Sets in P. D. E., (Lecture Notes in Mathematics, 1150, Springer-verlag, Berlin, Heidelberg, New York, Tokyo, 1985).CrossRefGoogle Scholar
[11]Krasnosel'skii, M.A. and Rutickii, Y.A., Convex Functions and Orlicz Spaces, (P. Noordhoff Ltd., Groningen, 1961).Google Scholar
[12]Migliaccio, L., “Sur une condition de Hardy-Littlewood-Polya”, C.R. Acad. Sci. Paris, Sér. I. Math. 297 (1983), 25–28.Google Scholar
[13]Moser, J., “A sharp form of an inequality by N. Trudinger”, Indiana Univ. Math. J., 20 (1971), 1077–1092.CrossRefGoogle Scholar
[14]Murthy, M.K.V. and Stampacchia, G., “Boundary value problems for some degenerate elliptic operators”, Ann. Math. Pura Appl., 90 (1971), 1–122.CrossRefGoogle Scholar
[15]Ryff, J., “Majorized functions and measure”, Indag. Math. 30 (1968), 431–437.CrossRefGoogle Scholar
[16]Stampacchia, G., “Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus”, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258.CrossRefGoogle Scholar
[17]Stampacchia, G., “Some limit cases of Lp-estimates for solutions of second order elliptic equations”, Comm. Pure Appl. Math. XVI (1963), 505–510.CrossRefGoogle Scholar
[18]Talenti, G., “Elliptic equations and rearrangements”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 697–718.Google Scholar
[19]Talenti, G., “Best constant in Sobolev inequality”, Ann. Mat. Pura Appl. 110 (1976), 353–372.CrossRefGoogle Scholar
[20]Trudinger, N.S., “On imbeddings into Orlicz spaces and some applications”, J. Math. Mech., 17 (1967), 473–484.Google Scholar
[21]Trudinger, N.S., “Linear elliptic operators with measurable coefficients”, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat., 27 (1973), 265–308.Google Scholar
[22]Weinberger, H.F., “Symmetrization in uniformly elliptic problems”, Studies in Math. Analysis and related topics, (Stanford Univ. Press, Stanford, California, 1962), 424–428.Google Scholar