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Orlicz capacities and applications to some existence questions for elliptic pdes having measure data

Published online by Cambridge University Press:  15 September 2003

Alberto Fiorenza  and
Alain Prignet
Alberto Fiorenza
Affiliation:
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, via Monteoliveto 3, 80134 Napoli, Italy, and Istituto per le Applicazioni del Calcolo “Mauro Picone", Sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino 111, 80131 Napoli, Italy; [email protected].
Alain Prignet
Affiliation:
Mathématiques, Université d'Orléans, rue de Chartres, 45067 Orléans Cedex 2, France; [email protected].
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Abstract

We study the sequence un, which is solution of $-{\rm div}(a(x,{\nabla}u_n)) + \Phi''(|u_n|)\,u_n= f_n+ g_n$ in Ω an open bounded set of RN and un= 0 on ∂Ω, when fn tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N-function Φ, and prove a non-existence result.

Keywords

Elliptic equationOrlicz spacemeasurecapacity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 317 - 341
Copyright
© EDP Sciences, SMAI, 2003

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References

D.R. Adams and L.I. Hedberg, Function spaces and potential theory. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 314 (1996).
Aissaoui, N., Bessel potentials in Orlicz spaces. Rev. Mat. Univ. Complut. Madrid 10 (1997) 55-79.
Aissaoui, N., Some developments of Strongly Nonlinear Potential Theory. Libertas Math. 19 (1999) 155-170.
Aissaoui, N. and Benkirane, A., Capacités dans les espaces d'Orlicz. Ann. Sci. Math. Québec 18 (1994) 1-23.
Baras, P. and Pierre, M., Singularités éliminables pour des équations semi-linéaires. Ann. Inst. Fourier (Grenoble) 34 (1984) 185-206. CrossRef
Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J.L., An L 1 theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995) 240-273.
Bénilan, P., Brezis, H. and Crandall, M., A semilinear elliptic equation in L 1(R N ). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1975) 523-555.
Boccardo, L. and Gallouët, T., Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17 (1992) 641-655. CrossRef
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.
G. Choquet, Theory of Capacities, Ann. Inst. Fourier (Grenoble) 5 (1953-1954) 131-295 (Ch. 1, Thm 4.1, p. 142).
Dal Maso, G., Murat, F., Orsina, L. and Prignet, A., Renormalized solutions for elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa CL. Sci. 28 (1999) 741-808.
Donaldson, T.K. and Trudinger, N.S., Orlicz-Sobolev spaces and embedding theorems. J. Funct. Anal. 8 (1971) 52-75. CrossRef
Fiorenza, A., An inequality for Jensen Means. Nonlinear Anal. 16 (1991) 191-198. CrossRef
Gallouët, T. and Morel, J.M., Resolution of a semilinear equation in L 1. Proc. Roy. Soc. Edinburgh 96 (1984) 275-288. CrossRef
Gustavsson, J. and Peetre, J., Interpolation of Orlicz spaces. Studia Math. 60 (1977) 33-59.
V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces. World Scientific (1991).
M.A. Krasnosel'skii and Ya.B. Rutickii, Convex functions and Orlicz Spaces. Noordhoff Ltd. (1961).
Leray, J. and Lions, J.-L., Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965) 97-107. CrossRef
L. Maligranda, Orlicz Spaces and Interpolation. Dep. de Matematica Univ. Estadual de Campinas, Campinas, Brazil (1989).
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).
J. Malý, D. Swanson and W.P. Ziemer, Fine behavior of functions with gradient in a Lorentz space (in preparation).
Maz'ja, V.G. and Havin, V.P., Nonlinear potential theory. Uspekhi Mat. Nauk 27 (1972) 67-138. English translation: Russian Math. Surveys 27 (1972) 71-148.
Orsina, L. and Prignet, A., Nonexistence of solutions for some nonlinear elliptic equations involving measures. Proc. Roy. Soc. Edinburgh Ser. A 130 (2000) 167-187. CrossRef
Persson, L.E., Interpolation with a parameter function. Math. Scand. 59 (1986) 199-222. CrossRef
M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces. Marcel Dekker (1991).
C.A. Rogers, Hausdorff Measures. Cambridge University Press (1970).
E.M. Stein, Singular Integrals and Differentiability properties of functions. Princeton University Press (1970).