Fine topology and quasilinear elliptic equations
Annales de l'Institut Fourier, Volume 39 (1989) no. 2, p. 293-318

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-2u)=0

continuous. Fine limits of quasiregular and BLD mappings are also studied.

Il est démontré que la topologie fine de type (1,p) définie à l’aide d’un critère de Wiener est la moins fine topologie rendant continues toutes les sursolutions de l’équation p-harmonique

div (|u|p-2u)=0.

Les limites fines d’applications quasi-régulières et de type BLD sont aussi étudiées.

@article{AIF_1989__39_2_293_0,
     author = {Heinonen, Juha and Kilpel\"ainen, Terro and Martio, Olli},
     title = {Fine topology and quasilinear elliptic equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {39},
     number = {2},
     year = {1989},
     pages = {293-318},
     doi = {10.5802/aif.1168},
     zbl = {0659.35038},
     mrnumber = {91b:31015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_2_293_0}
}
Heinonen, Juha; Kilpeläinen, Terro; Martio, Olli. Fine topology and quasilinear elliptic equations. Annales de l'Institut Fourier, Volume 39 (1989) no. 2, pp. 293-318. doi : 10.5802/aif.1168. http://www.numdam.org/item/AIF_1989__39_2_293_0/

[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. | MR 85c:31011 | Zbl 0545.31011

[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. | Numdam | MR 86h:31009 | Zbl 0545.31012

[AM] D. R. Adams and N. G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972), 169-197. | MR 47 #5272 | Zbl 0244.31012

[B] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., 175, Springer-Verlag, 1971. | MR 43 #7654 | Zbl 0222.31014

[D] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. | MR 85k:31001 | Zbl 0549.31001

[F1] B. Fuglede, The quasi topology associated with a countable subadditive set function, Ann. Inst. Fourier, Grenoble, 21-1, (1971), 123-169. | Numdam | Zbl 0197.19401

[F2] B. Fuglede, Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, Grenoble, 21-3 (1971), 227-244. | Numdam | MR 49 #9241 | Zbl 0208.13802

[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. | MR 82b:31006 | Zbl 0445.31003

[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. | MR 87e:30037 | Zbl 0607.31002

[GLM1] S. Granlund, P. Lindqvist and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc., 277 (1983), 43-73. | MR 84f:30030 | Zbl 0518.30024

[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. | MR 88a:31014 | Zbl 0633.31004

[HW] L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier, Grenoble, 33-4 (1983), 161-187. | Numdam | MR 85f:31015 | Zbl 0508.31008

[HK1] J. Heinonen and T. Kilpeläinen, A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat., 26 (1988), 87-105. | MR 89k:35079 | Zbl 0652.31006

[HK2] J. Heinonen and T. Kilpeläinen, Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. (to appear). | Zbl 0706.31015

[HK3] J. Heinonen and T. Kilpeläinen, On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc., 310 (1988), 239-255. | MR 89m:35091 | Zbl 0711.35052

[K] T. Kilpeläinen, Potential theory for supersolutions of degenerate elliptic equations (to appear). | Zbl 0688.31005

[L] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math., 365 (1986), 67-79. | MR 87e:31011 | Zbl 0572.31004

[LM] P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171. | MR 87g:35074 | Zbl 0607.35042

[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. | Numdam | MR 28 #4228 | Zbl 0116.30302

[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. | MR 89b:31001 | Zbl 0607.31001

[MRV1] O. Martio, S. Rickman and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 448 (1969), 1-40. | MR 41 #3756 | Zbl 0189.09204

[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. | Zbl 0197.05702

[MS] O. Martio and J. Sarvas, Density conditions in the n-capacity, Indiana Univ. Math. J., 26 (1977), 761-776. | MR 57 #16582 | Zbl 0365.30014

[MV] O. Martio and J. Väisälä, Elliptic equations and maps of bounded length distortion, Math. Ann., 282, (1988), 423-443. | MR 89m:35062 | Zbl 0632.35021

[M] N. G. Meyers, Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR 51 #3477 | Zbl 0334.31004

[R] Yu. G. Reshetnyak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh., 10 (1969), 1109-1138. (Russian). | MR 43 #2234 | Zbl 0199.20701