Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the Wiener criterion and quasilinear obstacle problems


Authors: Juha Heinonen and Tero Kilpeläinen
Journal: Trans. Amer. Math. Soc. 310 (1988), 239-255
MSC: Primary 35J85; Secondary 49A29
DOI: https://doi.org/10.1090/S0002-9947-1988-0965751-8
MathSciNet review: 965751
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 \vert\nabla u{\vert^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 [Enhancements On Off] (What's this?)

  • [AMS] 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 0180846 (31:5076)
  • [CK] L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Analyse Math. 37 (1980), 285-295. MR 583641 (82b:49009)
  • [D] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York and Berlin, 1984. MR 731258 (85k:31001)
  • [FZ] H. Federer and W. P. Ziemer, The Lebesgue set of a function whose partial derivatives are $ p$-th power summable, Indiana Univ. Math. J. 22 (1972), 139-158. MR 0435361 (55:8321)
  • [FM1] J. Frehse and U. Mosco, Sur la continuité ponctuelle des solutions locales faibles du problème d'obstacle, C. R. Acad. Sci. Paris Ser. A 295 (1982), 571-574. MR 685027 (84b:49011)
  • [FM2] -, Wiener obstacles, Séminaire Collège de France, Paris (H. Brezis and J. L. Lions, eds.), Vol. 6, Pitman, 1984. MR 772244 (86a:49009)
  • [GLM1] S. Granlund, P. Lindqvist and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), 43-73. MR 690040 (84f:30030)
  • [GLM2] -, Note on the $ PW$ $ B$-method in the non-linear case, Pacific J. Math. 125 (1986), 381-395.
  • [HW] L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161-187. MR 727526 (85f:31015)
  • [HK1] J. Heinonen and T. Kilpeläinen, $ A$-superharmonic functions and supersolutions of degenerate elliptic equations, Ark. Mat. (to appear). MR 948282 (89k:35079)
  • [HK2] -, Polar sets for supersolutions of degenerate elliptic equations, Math. Scand. (to appear). MR 994974 (90g:31007)
  • [KS] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. MR 567696 (81g:49013)
  • [LS] H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153-188. MR 0247551 (40:816)
  • [L] P. Lindqvist, On the definition and properties of $ p$-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79. MR 826152 (87e:31011)
  • [LM] P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171. MR 806413 (87g:35074)
  • [M1] V. G. Maz'ya, On the continuity at a boundary point of solutions of quasi-linear elliptic equations, Vestnik Leningrad Univ. Mat. Mech. Astronom. 13 (1970), 42-55; English transl., Vestnik Leningrad Univ. Math. 3 (1976), 225-242. MR 0274948 (43:706)
  • [M2] -, Sobolev spaces, Springer-Verlag, New York and Berlin, 1985. MR 817985 (87g:46056)
  • [MK] V. G. Maz'ya and V. P. Khavin, Non-linear potential theory, Russian Math. Surveys 27 (1972), 71-148.
  • [Me] N. G. Meyers, Continuity properties of potentials, Duke Math. J. 42 (1975), 157-166. MR 0367235 (51:3477)
  • [MZ] J. H. Michael and W. P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Analysis 10 (1986), 1427-1448. MR 869551 (88k:35083)
  • [Mo1] U. Mosco, Wiener criterion and potential estimates for the obstacle problem, Indiana Univ. Math. J. 36 (1987), 455-494. MR 905606 (88k:35084)
  • [Mo2] -, Wiener criteria and variational convergences, SFB 72, Preprint 797, Bonn, 1986.
  • [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 0276487 (43:2234)
  • [Sa] J. Sarvas, Symmetrization of condensers in $ n$-space, Ann. Acad. Sci. Fenn. Ser. A I Math. 522 (1972), 1-44. MR 0348108 (50:606)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J85, 49A29

Retrieve articles in all journals with MSC: 35J85, 49A29


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0965751-8
Keywords: Wiener criterion, obstacle problem, $ A$-superharmonic functions
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society