Fine topology and quasilinear elliptic equations

Heinonen, Juha; Kilpeläinen, Terro; Martio, Olli

Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 293-318.

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

$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=0.$$

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

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

$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=0$$

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

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

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

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

[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 | Zbl