On the Wiener criterion and quasilinear obstacle problems

Heinonen, Juha; Kilpeläinen, Tero

doi:10.1090/S0002-9947-1988-0965751-8

On the Wiener criterion and quasilinear obstacle problems
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by Juha Heinonen and Tero Kilpeläinen PDF

Trans. Amer. Math. Soc. 310 (1988), 239-255 Request permission

Abstract:

We study the Wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators $A$, $A(x, \nabla u) \cdot \nabla u \approx |\nabla u{|^p}$, in ${{\mathbf {R}}^n}$. Local solutions are continuous at Wiener points of the obstacle function; if $p > n - 1$, the converse is also shown to be true. If $p > n - 1$, then a characterization of the thinness of a set at a point is given in terms of $A$-superharmonic functions.

References

N. Aronszajn, F. Mulla, and P. Szeptycki, On spaces of potentials connected with $L^{p}$ classes, Ann. Inst. Fourier (Grenoble) 13 (1963), 211–306. MR 180846
L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Analyse Math. 37 (1980), 285–295. MR 583641, DOI 10.1007/BF02797689
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI 10.1512/iumj.1972.22.22013
Jens Frehse and Umberto Mosco, Sur la continuité ponctuelle des solutions locales faibles du problème d’obstacle, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 10, 571–574 (French, with English summary). MR 685027
J. Frehse and U. Mosco, Wiener obstacles, Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VI (Paris, 1982/1983) Res. Notes in Math., vol. 109, Pitman, Boston, MA, 1984, pp. 225–257. MR 772244
S. Granlund, P. Lindqvist, and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), no. 1, 43–73. MR 690040, DOI 10.1090/S0002-9947-1983-0690040-4

Note on the

method in the non-linear case

125

L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187. MR 727526
Juha Heinonen and Tero Kilpeläinen, $A$-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat. 26 (1988), no. 1, 87–105. MR 948282, DOI 10.1007/BF02386110
Juha Heinonen and Tero Kilpeläinen, Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. 63 (1988), no. 1, 136–150. MR 994974, DOI 10.7146/math.scand.a-12229
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
Hans Lewy and Guido Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153–188. MR 247551, DOI 10.1002/cpa.3160220203
Peter Lindqvist, On the definition and properties of $p$-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67–79. MR 826152, DOI 10.1515/crll.1986.365.67
P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171. MR 806413, DOI 10.1007/BF02392541
V. G. Maz′ja, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ. 25 (1970), no. 13, 42–55 (Russian, with English summary). MR 0274948
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3

Non-linear potential theory

27

Norman G. Meyers, Continuity properties of potentials, Duke Math. J. 42 (1975), 157–166. MR 367235
J. H. Michael and William P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), no. 12, 1427–1448. MR 869551, DOI 10.1016/0362-546X(86)90113-6
Umberto Mosco, Wiener criterion and potential estimates for the obstacle problem, Indiana Univ. Math. J. 36 (1987), no. 3, 455–494. MR 905606, DOI 10.1512/iumj.1987.36.36026

Wiener criteria and variational convergences

Ju. G. Rešetnjak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Ž. 10 (1969), 1109–1138 (Russian). MR 0276487
Jukka Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. A. I. 522 (1972), 44. MR 348108