| Title:
|
Problems involving $p$-Laplacian type equations and measures (English) |
| Author:
|
Kilpeläinen, Tero |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
243-250 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper I discuss two questions on $p$-Laplacian type operators: I characterize sets that are removable for Hölder continuous solutions and then discuss the problem of existence and uniqueness of solutions to $-\div (|\nabla u|^{p-2}\nabla u)=\mu $ with zero boundary values; here $\mu $ is a Radon measure. The joining link between the problems is the use of equations involving measures. (English) |
| Keyword:
|
$p$-Laplacian |
| Keyword:
|
removable sets |
| MSC:
|
35B60 |
| MSC:
|
35J60 |
| MSC:
|
35J70 |
| MSC:
|
35R05 |
| idZBL:
|
Zbl 1074.35536 |
| idMR:
|
MR1981529 |
| DOI:
|
10.21136/MB.2002.134164 |
| . |
| Date available:
|
2012-10-05T12:58:40Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134164 |
| . |
| Reference:
|
[1] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory.Springer, Berlin, 1995. MR 1411441 |
| Reference:
|
[2] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J. L.: An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), 241–273. MR 1354907 |
| Reference:
|
[3] Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures.Comm. Partial Diff. Eq. 17 (1992), 641–655. MR 1163440, 10.1080/03605309208820857 |
| Reference:
|
[4] Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data.Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 539–551. MR 1409661, 10.1016/S0294-1449(16)30113-5 |
| Reference:
|
[5] Carleson, L.: Selected Problems on Exceptional Sets.Van Nostrand, Princeton, N.Y., 1967. Zbl 0189.10903, MR 0225986 |
| Reference:
|
[6] Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data.Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 741–808. MR 1760541 |
| Reference:
|
[7] David, G., Mattila, P.: Removable sets for Lipschitz harmonic functions in the plane.Rev. Mat. Iberoam. 16 (2000), 137–215. MR 1768535, 10.4171/RMI/272 |
| Reference:
|
[8] Dolzmann, G., Hungerbühler, N., Müller, S.: Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right hand side.J. Reine Angew. Math. 520 (2000), 1–35. MR 1748270, 10.1515/crll.2000.022 |
| Reference:
|
[9] Greco, L., Iwaniec, T., Sbordone, C.: Inverting the $p$-harmonic operator.Manuscripta Math. 92 (1997), 249–258. MR 1428651, 10.1007/BF02678192 |
| Reference:
|
[10] Heinonen, J., Kilpeläinen, T.: $A$-superharmonic functions and supersolutions of degenerate elliptic equations.Ark. Mat. 26 (1988), 87–105. MR 0948282, 10.1007/BF02386110 |
| Reference:
|
[11] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.Oxford University Press, Oxford, 1993. MR 1207810 |
| Reference:
|
[12] Kilpeläinen, T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures.Potential Analysis 3 (1994), 265–272. 10.1007/BF01468246 |
| Reference:
|
[13] Kilpeläinen, T., Xu, X.: On the uniqueness problem for quasilinear elliptic equations involving measures.Rev. Mat. Iberoam. 12 (1996), 461–475. 10.4171/RMI/204 |
| Reference:
|
[14] Kilpeläinen, T., Zhong, X.: Removable sets for continuous solutions of quasilinear elliptic equations.(to appear). MR 1887015 |
| Reference:
|
[15] Lieberman, G.M.: Regularity of solutions to some degenerate double obstacle problems.Indiana Univ. Math. J. 40 (1991), 1009–1028. Zbl 0767.35029, MR 1129339, 10.1512/iumj.1991.40.40045 |
| Reference:
|
[16] Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures.Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes 104 (1996), 1–71. Zbl 0860.35041, MR 1386213 |
| Reference:
|
[17] Serrin, J.: Local behavior of solutions to quasi-linear equations.Acta Math. 111 (1964), 247–302. MR 0170096, 10.1007/BF02391014 |
| Reference:
|
[18] Serrin, J.: Pathological solutions of elliptic differential equations.Ann. Scuola Norm. Sup. Pisa 18 (1964), 385–387. Zbl 0142.37601, MR 0170094 |
| Reference:
|
[19] Trudinger, N., Wang, X. J.: On the weak continuity of elliptic operators and applications to potential theory.(to appear). MR 1890997 |
| Reference:
|
[20] Trudinger, N. S., Wang, X. J.: Dirichlet problems for quasilinear elliptic equations with measure data.Preprint. |
| Reference:
|
[21] Zhong, X.: On nonhomogeneous quasilinear elliptic equations.Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998). Zbl 0911.35048, MR 1648847 |
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