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

Fiorenza, Alberto; Prignet, Alain

doi:10.1051/cocv:2003015

Fiorenza, Alberto ; Prignet, Alain

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 317-341.

Résumé

We study the sequence $u_{n}$ , which is solution of $- div (a (x, 𝔻 u_{n})) + Φ^{''} (| u_{n} |) u_{n} = f_{n} + g_{n}$ in $Ω$ an open bounded set of $𝐑^{N}$ and $u_{n} = 0$ on $\partial Ω$ , 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 $Φ$ , and prove a non-existence result.

MR Zbl

DOI : 10.1051/cocv:2003015

Classification : 35J60, 46E30, 31C45
Mots-clés : elliptic equation, Orlicz space, measure, capacity

@article{COCV_2003__9__317_0,
     author = {Fiorenza, Alberto and Prignet, Alain},
     title = {Orlicz capacities and applications to some existence questions for elliptic {PDEs} having measure data},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {317--341},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003015},
     mrnumber = {1966536},
     zbl = {1075.35012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2003015/}
}

TY  - JOUR
AU  - Fiorenza, Alberto
AU  - Prignet, Alain
TI  - Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
SP  - 317
EP  - 341
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003015/
DO  - 10.1051/cocv:2003015
LA  - en
ID  - COCV_2003__9__317_0
ER  -

%0 Journal Article
%A Fiorenza, Alberto
%A Prignet, Alain
%T Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2003
%P 317-341
%V 9
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2003015/
%R 10.1051/cocv:2003015
%G en
%F COCV_2003__9__317_0

Fiorenza, Alberto; Prignet, Alain. Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 317-341. doi : 10.1051/cocv:2003015. http://www.numdam.org/articles/10.1051/cocv:2003015/

Bibliographie
Cité par

[1] D.R. Adams and L.I. Hedberg, Function spaces and potential theory. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 314 (1996). | MR | Zbl

[2] N. Aissaoui, Bessel potentials in Orlicz spaces. Rev. Mat. Univ. Complut. Madrid 10 (1997) 55-79. | MR | Zbl

[3] N. Aissaoui, Some developments of Strongly Nonlinear Potential Theory. Libertas Math. 19 (1999) 155-170. | MR | Zbl

[4] N. Aissaoui and A. Benkirane, Capacités dans les espaces d'Orlicz. Ann. Sci. Math. Québec 18 (1994) 1-23. | Zbl

[5] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires. Ann. Inst. Fourier (Grenoble) 34 (1984) 185-206. | Numdam | MR | Zbl

[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 | Zbl

[7] P. Bénilan, H. Brezis and M. Crandall, A semilinear elliptic equation in $L^{1} (𝐑^{N})$ . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1975) 523-555. | Numdam | MR | Zbl

[8] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17 (1992) 641-655. | MR | Zbl

[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 | Zbl

[10] G. Choquet, Theory of Capacities, Ann. Inst. Fourier (Grenoble) 5 (1953-1954) 131-295 (Ch. 1, Thm 4.1, p. 142). | EuDML | Numdam | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[12] T.K. Donaldson and N.S. Trudinger, Orlicz-Sobolev spaces and embedding theorems. J. Funct. Anal. 8 (1971) 52-75. | MR | Zbl

[13] A. Fiorenza, An inequality for Jensen Means. Nonlinear Anal. 16 (1991) 191-198. | MR | Zbl

[14] T. Gallouët and J.M. Morel, Resolution of a semilinear equation in $L^{1}$ . Proc. Roy. Soc. Edinburgh 96 (1984) 275-288. | MR | Zbl

[15] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces. Studia Math. 60 (1977) 33-59. | EuDML | MR | Zbl

[16] V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces. World Scientific (1991). | MR | Zbl

[17] M.A. Krasnosel'Skii and Ya.B. Rutickii, Convex functions and Orlicz Spaces. Noordhoff Ltd. (1961). | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[19] L. Maligranda, Orlicz Spaces and Interpolation. Dep. de Matematica Univ. Estadual de Campinas, Campinas, Brazil (1989). | MR | Zbl

[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 | Zbl

[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. | MR | Zbl

[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 | Zbl

[24] L.E. Persson, Interpolation with a parameter function. Math. Scand. 59 (1986) 199-222. | EuDML | MR | Zbl

[25] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces. Marcel Dekker (1991). | MR | Zbl

[26] C.A. Rogers, Hausdorff Measures. Cambridge University Press (1970). | MR | Zbl

[27] E.M. Stein, Singular Integrals and Differentiability properties of functions. Princeton University Press (1970). | MR | Zbl

Cité par Sources :