no. 1
Capacitary strong type estimates in semilinear problems
Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 117-135.

Nous montrons l’équivalence de diverses estimations capacitaires de type fort. Certaines d’entre elles apparaissent dans la caractérisation des mesures μ qui sont admissibles pour l’existence de solutions de problèmes elliptiques semi-linéaires avec croissance polynomiale. D’autres sont bien connues comme caractérisant les mesures μ telles que l’espace de Sobolev W 2,p s’injecte continûment dans L p (μ). La motivation vient essentiellement des problèmes semilinéaires : des descriptions très simples des données admissibles peuvent être ainsi données. La démonstration utilise de façon assez surprenenante la théorie des intégrales singulières avec poids de type A p .

We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures μ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures μ for which the Sobolev space W 2,p can be imbedded into L p (μ). The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular integrals with A p -weights.

@article{AIF_1991__41_1_117_0,
     author = {Adams, D. and Pierre, Michel},
     title = {Capacitary strong type estimates in semilinear problems},
     journal = {Annales de l'Institut Fourier},
     pages = {117--135},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {1},
     year = {1991},
     doi = {10.5802/aif.1251},
     zbl = {0741.35012},
     mrnumber = {92m:35074},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1251/}
}
TY  - JOUR
AU  - Adams, D.
AU  - Pierre, Michel
TI  - Capacitary strong type estimates in semilinear problems
JO  - Annales de l'Institut Fourier
PY  - 1991
SP  - 117
EP  - 135
VL  - 41
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1251/
DO  - 10.5802/aif.1251
LA  - en
ID  - AIF_1991__41_1_117_0
ER  - 
%0 Journal Article
%A Adams, D.
%A Pierre, Michel
%T Capacitary strong type estimates in semilinear problems
%J Annales de l'Institut Fourier
%D 1991
%P 117-135
%V 41
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1251/
%R 10.5802/aif.1251
%G en
%F AIF_1991__41_1_117_0
Adams, D.; Pierre, Michel. Capacitary strong type estimates in semilinear problems. Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 117-135. doi : 10.5802/aif.1251. https://aif.centre-mersenne.org/articles/10.5802/aif.1251/

[1] D.R. Adams, On the existence of capacitary strong types estimates in RN, Arkiv för Matematik, Vol. 14, n° 1 (1976), 125-140 | MR | Zbl

[2] D.R. Adams, Lectures on Lp-potential theory, Umea Univ. report, 2 (1981)

[3] D.R. Adams, J.C. Polking, The equivalence of two definitions of capacity, Proc. of A.M.S., 37 (1973), 529-534. | MR | Zbl

[4] P. Baras, M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier, Grenoble, 34-1 (1984), 185-206. | Numdam | MR | Zbl

[5] P. Baras, M. Pierre, Critère d'existence des solutions positives pour des équations semi-linéaires non monotones, Ann. I.H.P., 2 n° 3 (1985), 185-212 | Numdam | MR | Zbl

[6] R.R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Mathematica, 51 (1974), 241-250. | MR | Zbl

[7] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag (1983), 2nd édition. | MR | Zbl

[8] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102. | MR | Zbl

[9] L.I. Hedberg, On certain convolution inequalities, Proc. of A.M.S., 36, n° 2 (1972), 505-510. | MR | Zbl

[10] V.G. Maz'Ya, On some integral inequalities for functions of several variables, Problems in Math. Analysis, Leningrad n° 3 (1973), (Russian).

[11] V.G. Maz'Ya, T.O. Shaposhnikova, Theory of multipliers spaces of differentiable functions, Monographes and Studies in Math., 23, Pitman. | MR | Zbl

[12] N.G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand., 26 (1970), 255-292 | MR | Zbl

[13] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans, A.M.S., 165 (1972), 207-226. | MR | Zbl

[14] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Pisa, 13 (1959), 115-162. | Numdam | MR | Zbl

[15] M. Pierre, Problèmes semi-linéaires avec données mesures, Séminaires Goulaouic-Schwartz, Ecole Polytechnique, exposé n° XIII, 1983. | Numdam | MR | Zbl

[16] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton N.J. (1970). | MR | Zbl

Cité par Sources :