×

Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data. (English) Zbl 1075.35012

Summary: We study the sequence \(u_n \), which is solution of \(-\text{div} (a(x,\nabla u_nn )) + \Phi ''(| u_n | )\,u_n = f_n + g_n \) in \(\Omega \) an open bounded set of \(\mathbb{R}^N\) and \(u_n = 0\) on \(\partial \Omega \), when \(f_n\) tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the \(N\)-function \(\Phi \), and prove a non-existence result.

MSC:

35J60 Nonlinear elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
31C45 Other generalizations (nonlinear potential theory, etc.)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] D.R. Adams and L.I. Hedberg , Function spaces and potential theory . Springer-Verlag, Berlin, Grundlehren Math. Wiss. 314 ( 1996 ). MR 1411441 | Zbl 0834.46021 · Zbl 0834.46021
[2] N. Aissaoui , Bessel potentials in Orlicz spaces . Rev. Mat. Univ. Complut. Madrid 10 ( 1997 ) 55 - 79 . MR 1452563 | Zbl 0899.46019 · Zbl 0899.46019
[3] N. Aissaoui , Some developments of Strongly Nonlinear Potential Theory . Libertas Math. 19 ( 1999 ) 155 - 170 . MR 1726166 | Zbl 0976.31007 · Zbl 0976.31007
[4] N. Aissaoui and A. Benkirane , Capacités dans les espaces d’Orlicz . Ann. Sci. Math. Québec 18 ( 1994 ) 1 - 23 . Zbl 0822.31006 · Zbl 0822.31006
[5] P. Baras and M. Pierre , Singularités éliminables pour des équations semi-linéaires . Ann. Inst. Fourier (Grenoble) 34 ( 1984 ) 185 - 206 . Numdam | MR 743627 | Zbl 0519.35002 · Zbl 0519.35002
[6] P. Bénilan , L. Boccardo , T. Gallouët , R. Gariepy , M. Pierre and J.L. Vazquez , An \(L^1\) theory of existence and uniqueness of nonlinear elliptic equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 ( 1995 ) 240 - 273 . Numdam | MR 1354907 | Zbl 0866.35037 · Zbl 0866.35037
[7] P. Bénilan , H. Brezis and M. Crandall , A semilinear elliptic equation in \(L^1({{\mathbf{R}}^N})\) . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 ( 1975 ) 523 - 555 . Numdam | MR 390473 | Zbl 0314.35077 · Zbl 0314.35077
[8] L. Boccardo and T. Gallouët , Nonlinear elliptic equations with right-hand side measures . Comm. Partial Differential Equations 17 ( 1992 ) 641 - 655 . MR 1163440 | Zbl 0812.35043 · Zbl 0812.35043
[9] H. Brezis , Nonlinear elliptic equations involving measures , in Contributions to nonlinear partial differential equations (Madrid, 1981). Pitman, Boston, Mass.-London, Res. Notes in Math. 89 1983) 82 - 89 . MR 730798 | Zbl 0533.35038 · Zbl 0533.35038
[10] G. Choquet , Theory of Capacities , Ann. Inst. Fourier (Grenoble) 5 ( 1953 - 1954 ) 131 - 295 (Ch. 1, Thm 4.1, p. 142). Numdam | MR 80760 | Zbl 0064.35101 · Zbl 0064.35101
[11] G. Dal Maso , F. Murat , L. Orsina and A. Prignet , Renormalized solutions for elliptic equations with general measure data . Ann. Scuola Norm. Sup. Pisa CL. Sci. 28 ( 1999 ) 741 - 808 . Numdam | MR 1760541 | Zbl 0958.35045 · Zbl 0958.35045
[12] T.K. Donaldson and N.S. Trudinger , Orlicz-Sobolev spaces and embedding theorems . J. Funct. Anal. 8 ( 1971 ) 52 - 75 . Zbl 0216.15702 · Zbl 0216.15702
[13] A. Fiorenza , An inequality for Jensen Means . Nonlinear Anal. 16 ( 1991 ) 191 - 198 . MR 1090790 | Zbl 0737.46009 · Zbl 0737.46009
[14] T. Gallouët and J.M. Morel , Resolution of a semilinear equation in \(L^1\) . Proc. Roy. Soc. Edinburgh 96 ( 1984 ) 275 - 288 . MR 760776 | Zbl 0573.35030 · Zbl 0573.35030
[15] J. Gustavsson and J. Peetre , Interpolation of Orlicz spaces . Studia Math. 60 ( 1977 ) 33 - 59 . MR 438102 | Zbl 0353.46019 · Zbl 0353.46019
[16] V. Kokilashvili and M. Krbec , Weighted inequalities in Lorentz and Orlicz spaces . World Scientific ( 1991 ). MR 1156767 | Zbl 0751.46021 · Zbl 0751.46021
[17] M.A. Krasnosel’skii and Ya.B. Rutickii , Convex functions and Orlicz Spaces . Noordhoff Ltd. ( 1961 ). Zbl 0095.09103 · Zbl 0095.09103
[18] J. Leray and J.-L. Lions , Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder . Bull. Soc. Math. France 93 ( 1965 ) 97 - 107 . Numdam | Zbl 0132.10502 · Zbl 0132.10502
[19] L. Maligranda , Orlicz Spaces and Interpolation . Dep. de Matematica Univ. Estadual de Campinas, Campinas, Brazil ( 1989 ). MR 2264389 | Zbl 0874.46022 · Zbl 0874.46022
[20] J. Malý , Coarea properties of Sobolev functions , in Proc. Function Spaces, Differential Operators and Nonlinear Analysis (The Hans Triebel Anniversary Volume). Birkhäuser, Basel (to appear). MR 1984185 | Zbl 1036.46025 · Zbl 1036.46025
[21] J. Malý , D. Swanson and W.P. Ziemer , Fine behavior of functions with gradient in a Lorentz space (in preparation).
[22] V.G. Maz’ja and V.P. Havin , Nonlinear potential theory . Uspekhi Mat. Nauk 27 ( 1972 ) 67 - 138 . English translation: Russian Math. Surveys 27 ( 1972 ) 71 - 148 . Zbl 0269.31004 · Zbl 0269.31004
[23] L. Orsina and A. Prignet , Nonexistence of solutions for some nonlinear elliptic equations involving measures . Proc. Roy. Soc. Edinburgh Ser. A 130 ( 2000 ) 167 - 187 . MR 1742585 | Zbl 0953.35048 · Zbl 0953.35048
[24] L.E. Persson , Interpolation with a parameter function . Math. Scand. 59 ( 1986 ) 199 - 222 . MR 884656 | Zbl 0619.46064 · Zbl 0619.46064
[25] M.M. Rao and Z.D. Ren , Theory of Orlicz Spaces . Marcel Dekker ( 1991 ). MR 1113700 | Zbl 0724.46032 · Zbl 0724.46032
[26] C.A. Rogers , Hausdorff Measures . Cambridge University Press ( 1970 ). MR 281862 | Zbl 0204.37601 · Zbl 0204.37601
[27] E.M. Stein , Singular Integrals and Differentiability properties of functions . Princeton University Press ( 1970 ). MR 290095 | Zbl 0207.13501 · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.