It is shown that the -fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the -Laplace equation
continuous. Fine limits of quasiregular and BLD mappings are also studied.
Il est démontré que la topologie fine de type définie à l’aide d’un critère de Wiener est la moins fine topologie rendant continues toutes les sursolutions de l’équation -harmonique
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}, pages = {293--318}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {39}, number = {2}, year = {1989}, doi = {10.5802/aif.1168}, zbl = {0659.35038}, mrnumber = {91b:31015}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1168/} }
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 SP - 293 EP - 318 VL - 39 IS - 2 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1168/ DO - 10.5802/aif.1168 LA - en ID - AIF_1989__39_2_293_0 ER -
%0 Journal Article %A Heinonen, Juha %A Kilpeläinen, Terro %A Martio, Olli %T Fine topology and quasilinear elliptic equations %J Annales de l'Institut Fourier %D 1989 %P 293-318 %V 39 %N 2 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1168/ %R 10.5802/aif.1168 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1168/
[AH] Inclusion relations among fine topologies in non-linear potential theory, Indiana Univ. Math. J., 33 (1984), 117-126. | MR | Zbl
and ,[AL] Fine and quasi connectedness in nonlinear potential theory, Ann. Inst. Fourier, Grenoble, 35-1 (1985), 57-73. | EuDML | Numdam | MR | Zbl
and ,[AM] Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972), 169-197. | MR | Zbl
and ,[B] On topologies and boundaries in potential theory, Lecture Notes in Math., 175, Springer-Verlag, 1971. | MR | Zbl
,[D] Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. | MR | Zbl
,[F1] The quasi topology associated with a countable subadditive set function, Ann. Inst. Fourier, Grenoble, 21-1, (1971), 123-169. | EuDML | Numdam | Zbl
,[F2] Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, Grenoble, 21-3 (1971), 227-244. | EuDML | Numdam | MR | Zbl
,[F3] 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 | Zbl
,[F4] 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 | Zbl
,[GLM1] Conformally invariant variational integrals, Trans. Amer. Math. Soc., 277 (1983), 43-73. | MR | Zbl
, and ,[GLM2] Note on the PWB-method in the non-linear case, Pacific J. Math., 125 (1986), 381-395. | MR | Zbl
, and ,[HW] Thin sets in nonlinear potential theory, Ann. Inst. Fourier, Grenoble, 33-4 (1983), 161-187. | Numdam | MR | Zbl
and ,[HK1] A-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat., 26 (1988), 87-105. | MR | Zbl
and ,[HK2] Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. (to appear). | Zbl
and ,[HK3] On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc., 310 (1988), 239-255. | MR | Zbl
and ,[K] Potential theory for supersolutions of degenerate elliptic equations (to appear). | Zbl
,[L] On the definition and properties of p-superharmonic functions, J. Reine Angew. Math., 365 (1986), 67-79. | MR | Zbl
,[LM] Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math., 155 (1985), 153-171. | MR | Zbl
and ,[LSW] Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (III), 17 (1963), 43-77. | Numdam | MR | Zbl
, and ,[LMZ] Fine topology methods in real analysis and potential theory, Lecture Notes in Math., 1189, Springer-Verlag, 1986. | MR | Zbl
, and ,[MRV1] Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 448 (1969), 1-40. | MR | Zbl
, and ,[MRV2] Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 464 (1970), 1-13. | Zbl
, and ,[MS] Density conditions in the n-capacity, Indiana Univ. Math. J., 26 (1977), 761-776. | MR | Zbl
and ,[MV] Elliptic equations and maps of bounded length distortion, Math. Ann., 282, (1988), 423-443. | MR | Zbl
and ,[M] Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR | Zbl
,[R] The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh., 10 (1969), 1109-1138. (Russian). | MR | Zbl
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