# Article

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Keywords:
$p$-Laplacian; removable sets
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.
References:
[1] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Springer, Berlin, 1995. MR 1411441
[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
[3] Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures. Comm. Partial Diff. Eq. 17 (1992), 641–655. MR 1163440
[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
[5] Carleson, L.: Selected Problems on Exceptional Sets. Van Nostrand, Princeton, N.Y., 1967. MR 0225986 | Zbl 0189.10903
[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
[7] David, G., Mattila, P.: Removable sets for Lipschitz harmonic functions in the plane. Rev. Mat. Iberoam. 16 (2000), 137–215. MR 1768535
[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
[9] Greco, L., Iwaniec, T., Sbordone, C.: Inverting the $p$-harmonic operator. Manuscripta Math. 92 (1997), 249–258. MR 1428651
[10] Heinonen, J., Kilpeläinen, T.: $A$-superharmonic functions and supersolutions of degenerate elliptic equations. Ark. Mat. 26 (1988), 87–105. MR 0948282
[11] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford, 1993. MR 1207810
[12] Kilpeläinen, T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Analysis 3 (1994), 265–272.
[13] Kilpeläinen, T., Xu, X.: On the uniqueness problem for quasilinear elliptic equations involving measures. Rev. Mat. Iberoam. 12 (1996), 461–475.
[14] Kilpeläinen, T., Zhong, X.: Removable sets for continuous solutions of quasilinear elliptic equations. (to appear). MR 1887015
[15] Lieberman, G.M.: Regularity of solutions to some degenerate double obstacle problems. Indiana Univ. Math. J. 40 (1991), 1009–1028. MR 1129339 | Zbl 0767.35029
[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. MR 1386213 | Zbl 0860.35041
[17] Serrin, J.: Local behavior of solutions to quasi-linear equations. Acta Math. 111 (1964), 247–302. MR 0170096
[18] Serrin, J.: Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa 18 (1964), 385–387. MR 0170094 | Zbl 0142.37601
[19] Trudinger, N., Wang, X. J.: On the weak continuity of elliptic operators and applications to potential theory. (to appear). MR 1890997
[20] Trudinger, N. S., Wang, X. J.: Dirichlet problems for quasilinear elliptic equations with measure data. Preprint.
[21] Zhong, X.: On nonhomogeneous quasilinear elliptic equations. Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998). MR 1648847 | Zbl 0911.35048

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