Capacitary strong type estimates in semilinear problems

D. Adams; Michel Pierre

Annales de l'institut Fourier (1991)

  • Volume: 41, Issue: 1, page 117-135
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Adams, D., and Pierre, Michel. "Capacitary strong type estimates in semilinear problems." Annales de l'institut Fourier 41.1 (1991): 117-135. <http://eudml.org/doc/74909>.

@article{Adams1991,
abstract = {We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu $ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu $ for which the Sobolev space $W^\{2,p\}$ can be imbedded into $L^p(\mu )$. 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.},
author = {Adams, D., Pierre, Michel},
journal = {Annales de l'institut Fourier},
keywords = {Sobolev spaces; removable sets; existence; class of weak solutions},
language = {eng},
number = {1},
pages = {117-135},
publisher = {Association des Annales de l'Institut Fourier},
title = {Capacitary strong type estimates in semilinear problems},
url = {http://eudml.org/doc/74909},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Adams, D.
AU - Pierre, Michel
TI - Capacitary strong type estimates in semilinear problems
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 1
SP - 117
EP - 135
AB - We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu $ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu $ for which the Sobolev space $W^{2,p}$ can be imbedded into $L^p(\mu )$. 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.
LA - eng
KW - Sobolev spaces; removable sets; existence; class of weak solutions
UR - http://eudml.org/doc/74909
ER -

References

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  9. [9] L.I. HEDBERG, On certain convolution inequalities, Proc. of A.M.S., 36, n° 2 (1972), 505-510. Zbl0283.26003MR47 #794
  10. [10] V.G. MAZ'YA, On some integral inequalities for functions of several variables, Problems in Math. Analysis, Leningrad n° 3 (1973), (Russian). 
  11. [11] V.G. MAZ'YA, T.O. SHAPOSHNIKOVA, Theory of multipliers spaces of differentiable functions, Monographes and Studies in Math., 23, Pitman. Zbl0645.46031MR87j:46074
  12. [12] N.G. MEYERS, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand., 26 (1970), 255-292 Zbl0242.31006MR43 #3474
  13. [13] B. MUCKENHOUPT, Weighted norm inequalities for the Hardy maximal function, Trans, A.M.S., 165 (1972), 207-226. Zbl0236.26016MR45 #2461
  14. [14] L. NIRENBERG, On elliptic partial differential equations, Ann. Scuola Norm. Pisa, 13 (1959), 115-162. Zbl0088.07601MR22 #823
  15. [15] M. PIERRE, Problèmes semi-linéaires avec données mesures, Séminaires Goulaouic-Schwartz, Ecole Polytechnique, exposé n° XIII, 1983. Zbl0533.35039MR85k:35089
  16. [16] E.M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton N.J. (1970). Zbl0207.13501MR44 #7280

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