Fine topology and quasilinear elliptic equations

Juha Heinonen; Terro Kilpeläinen; Olli Martio

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 2, page 293-318
  • ISSN: 0373-0956

Abstract

top
It is shown that the ( 1 , p ) -fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the p -Laplace equation div ( | ∇ u | p - 2 ∇ u ) = 0 continuous. Fine limits of quasiregular and BLD mappings are also studied.

How to cite

top

Heinonen, Juha, Kilpeläinen, Terro, and Martio, Olli. "Fine topology and quasilinear elliptic equations." Annales de l'institut Fourier 39.2 (1989): 293-318. <http://eudml.org/doc/74832>.

@article{Heinonen1989,
abstract = {It is shown that the $(1,p)$-fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the $p$-Laplace equation\begin\{\}\{\rm div\}\,(\vert \nabla u\vert ^\{p-2\} \nabla u)=0\end\{\}continuous. Fine limits of quasiregular and BLD mappings are also studied.},
author = {Heinonen, Juha, Kilpeläinen, Terro, Martio, Olli},
journal = {Annales de l'institut Fourier},
keywords = {quasilinear; fine topology; Wiener criterion; supersolutions; p-Laplace equation; continuous; Fine limits},
language = {eng},
number = {2},
pages = {293-318},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fine topology and quasilinear elliptic equations},
url = {http://eudml.org/doc/74832},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Heinonen, Juha
AU - Kilpeläinen, Terro
AU - Martio, Olli
TI - Fine topology and quasilinear elliptic equations
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 2
SP - 293
EP - 318
AB - It is shown that the $(1,p)$-fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the $p$-Laplace equation\begin{}{\rm div}\,(\vert \nabla u\vert ^{p-2} \nabla u)=0\end{}continuous. Fine limits of quasiregular and BLD mappings are also studied.
LA - eng
KW - quasilinear; fine topology; Wiener criterion; supersolutions; p-Laplace equation; continuous; Fine limits
UR - http://eudml.org/doc/74832
ER -

References

top
  1. [AH] D. R. ADAMS and L. I. HEDBERG, Inclusion relations among fine topologies in non-linear potential theory, Indiana Univ. Math. J., 33 (1984), 117-126. Zbl0545.31011MR85c:31011
  2. [AL] D. R. ADAMS and J. L. LEWIS, Fine and quasi connectedness in nonlinear potential theory, Ann. Inst. Fourier, Grenoble, 35-1 (1985), 57-73. Zbl0545.31012MR86h:31009
  3. [AM] D. R. ADAMS and N. G. MEYERS, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972), 169-197. Zbl0244.31012MR47 #5272
  4. [B] M. BRELOT, On topologies and boundaries in potential theory, Lecture Notes in Math., 175, Springer-Verlag, 1971. Zbl0222.31014MR43 #7654
  5. [D] J. L. DOOB, Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. Zbl0549.31001MR85k:31001
  6. [F1] B. FUGLEDE, The quasi topology associated with a countable subadditive set function, Ann. Inst. Fourier, Grenoble, 21-1, (1971), 123-169. Zbl0197.19401
  7. [F2] B. FUGLEDE, Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, Grenoble, 21-3 (1971), 227-244. Zbl0208.13802MR49 #9241
  8. [F3] B. FUGLEDE, Asymptotic paths for subharmonic functions and polygonal connectedness of fine domains, Séminaire de Théorie du Potentiel, Paris, n° 5, Lecture Notes in Math., 814, Springer-Verlag, 1980, pp. 97-116. Zbl0445.31003MR82b:31006
  9. [F4] B. FUGLEDE, Value distribution of harmonic and finely harmonic morphisms and applications in complex analysis, Ann. Acad. Sci. Fenn. Ser. A I Math., 11 (1986), 111-135. Zbl0607.31002MR87e:30037
  10. [GLM1] S. GRANLUND, P. LINDQVIST and O. MARTIO, Conformally invariant variational integrals, Trans. Amer. Math. Soc., 277 (1983), 43-73. Zbl0518.30024MR84f:30030
  11. [GLM2] S. GRANLUND, P. LINDQVIST and O. MARTIO, Note on the PWB-method in the non-linear case, Pacific J. Math., 125 (1986), 381-395. Zbl0633.31004MR88a:31014
  12. [HW] L. I. HEDBERG and Th. H. WOLFF, Thin sets in nonlinear potential theory, Ann. Inst. Fourier, Grenoble, 33-4 (1983), 161-187. Zbl0508.31008MR85f:31015
  13. [HK1] J. HEINONEN and T. KILPELÄINEN, A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat., 26 (1988), 87-105. Zbl0652.31006MR89k:35079
  14. [HK2] J. HEINONEN and T. KILPELÄINEN, Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. (to appear). Zbl0706.31015
  15. [HK3] J. HEINONEN and T. KILPELÄINEN, On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc., 310 (1988), 239-255. Zbl0711.35052MR89m:35091
  16. [K] T. KILPELÄINEN, Potential theory for supersolutions of degenerate elliptic equations (to appear). Zbl0688.31005
  17. [L] P. LINDQVIST, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math., 365 (1986), 67-79. Zbl0572.31004MR87e:31011
  18. [LM] P. LINDQVIST and O. MARTIO, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171. Zbl0607.35042MR87g:35074
  19. [LSW] W. LITTMAN, G. STAMPACCHIA and H. F. WEINBERGER, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (III), 17 (1963), 43-77. Zbl0116.30302MR28 #4228
  20. [LMZ] J. LUKEŠ, J. MALÝ and L. ZAJÍčEK, Fine topology methods in real analysis and potential theory, Lecture Notes in Math., 1189, Springer-Verlag, 1986. Zbl0607.31001MR89b:31001
  21. [MRV1] O. MARTIO, S. RICKMAN and J. VÄISÄLÄ, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 448 (1969), 1-40. Zbl0189.09204MR41 #3756
  22. [MRV2] O. MARTIO, S. RICKMAN and J. VÄISÄLÄ, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 464 (1970), 1-13. Zbl0197.05702
  23. [MS] O. MARTIO and J. SARVAS, Density conditions in the n-capacity, Indiana Univ. Math. J., 26 (1977), 761-776. Zbl0365.30014MR57 #16582
  24. [MV] O. MARTIO and J. VÄISÄLÄ, Elliptic equations and maps of bounded length distortion, Math. Ann., 282, (1988), 423-443. Zbl0632.35021MR89m:35062
  25. [M] N. G. MEYERS, Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. Zbl0334.31004MR51 #3477
  26. [R] Yu. G. RESHETNYAK, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh., 10 (1969), 1109-1138. (Russian). Zbl0199.20701MR43 #2234

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.