Skip to main content
SpringerLink
Log in

Wiener regularity for large solutions of nonlinear equations

  • Published:
Arkiv för Matematik

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Adams, D. R., Sets and functions of finiteL p-capacity,Indiana Univ. Math. J. 27 (1978), 611–627.

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, D. R.,L p potential theory techniques and nonlinear PDE, inPotential Theory (Nagoya, 1990) (Kishi, M., ed.), pp. 1–15, de Gruyter, Berlin, 1992.

    Google Scholar 

  3. Adams, D. R., Potential and capacity before and after Wiener, inProceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, Mich., 1994) (Mandrekar, V. and Masani, P. R., eds.), pp. 63–83, Proc. Sympos. Appl. Math.52, Amer. Math. Soc., Providence, R. I., 1997.

    Google Scholar 

  4. Adams, D. R. andHeard, A., The necessity of the Wiener test for some semi-linear elliptic equations,Indiana Univ. Math. J. 41 (1992), 109–124.

    MathSciNet  Google Scholar 

  5. Adams, D. R. andHedberg, L. I.,Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg, 1996.

    Google Scholar 

  6. Adams, D. R. andPierre, M., Capacitary strong type estimates in semilinear problems,Ann. Inst. Fourier (Grenoble) 41 (1991), 117–135.

    MathSciNet  Google Scholar 

  7. Adams, D. R. andPolking, J. C., The equivalence of two definitions of capacity,Proc. Amer. Math. Soc. 37 (1973), 529–534.

    MathSciNet  Google Scholar 

  8. Baras, P. andPierre, M., Singularités éliminables pour des équations semi-linéaires,Ann. Inst. Fourier (Grenoble) 34:1 (1984), 185–206.

    MathSciNet  Google Scholar 

  9. Bauman, P., A Wiener test for nondivergence structure, second-order elliptic equations,Indiana Univ. Math. J. 34 (1985), 825–844.

    Article  MATH  MathSciNet  Google Scholar 

  10. Brelot, M.,Lectures on Potential Theory, Tata Institute of Fundamental Research, Bombay, 1967.

    Google Scholar 

  11. Brezis, M. andVéron, L., Removable singularities for some nonlinear elliptic equations,Arch. Rational Mech. Anal. 75 (1980/81), 1–6.

    Article  MathSciNet  Google Scholar 

  12. Carleson, L.,Selected Problems on Exceptional Sets, Van Nostrand, Princeton, N. J., 1967.

    Google Scholar 

  13. Dal Maso, G. andMosco, U., Wiener criteria end energy decay for relaxed Dirichlet problems,Arch. Rational Mech. Anal. 95 (1986), 345–387.

    MathSciNet  Google Scholar 

  14. Delanoë, P., Generalized stereographic projections with prescribed scalar curvature, inGeometry and Nonlinear Partial Differential Equations (Fayetteville, Ark., 1990), (Oliker, V. and Treibergs, A., eds.), Contemp. Math.127, pp. 17–25, Amer. Math. Soc., Providence, R. I., 1992.

    Google Scholar 

  15. Delmas, J.-F. andDhersin, J.-S., Characterization ofG-regularity for super-Brownian motion and consequences for parabolic partial differential equations,Ann. Probab. 27 (1999), 731–750.

    Article  MathSciNet  Google Scholar 

  16. Delmas, J.-F. andDhersin, J.-S., Kolmogorov’s test for the Brownian snake,Ann. Probab. 29 (2001), 305–316.

    MathSciNet  Google Scholar 

  17. Dhersin, J.-S. andLe Gall, J.-F., Kiener’s test for super-Brownian motion and the Brownian snake,Probab. Theory Related Fields 108 (1997), 103–129.

    Article  MathSciNet  Google Scholar 

  18. Dhersin, J.-S. andLe Gall, J.-F., Kolmogorov’s test for super-Brownian motion,Ann. Probab. 26 (1998), 1041–1056.

    MathSciNet  Google Scholar 

  19. Dynkin, E. B., A probabilistic approach to one class of nonlinear differential equations,Probab. Theory Related Fields 89 (1991), 89–115.

    Article  MATH  MathSciNet  Google Scholar 

  20. Dynkin, E. B.,An Introduction to Branching Measure-valued Processes, Amer. Math. Soc., Providence, R. I., 1994.

    Google Scholar 

  21. Dynkin, E. B.,Diffusions, Superdiffusions and Partial Differential Equations, Amer. Math. Soc. Colloq. Publ.50, Amer. Math. Soc., Providence, R. I., 2002.

    Google Scholar 

  22. Dynkin, E. B. andKuznetsov, S. E., superdiffusions and removable singularities for quasilinear partial differential equations,Comm. Pure Appl. Math. 49 (1996), 125–176.

    Article  MathSciNet  Google Scholar 

  23. Dynkin, E. B. andKuznetsov, S. E., Fine topology and fine trace on the boundary associated with a class of semilinear differential equations,Comm. Pure Appl. Math. 51 (1998), 897–936.

    Article  MathSciNet  Google Scholar 

  24. Etheridge, A. M.,An Introduction to Superprocesses, Amer. Math. Soc., Providence, R. I., 2000.

    Google Scholar 

  25. Evans, L. C. andGariepy, R. F., Wiener’s criterion for the heat equation,Arch. Rational Mech. Anal. 78 (1982), 293–314.

    Article  MathSciNet  Google Scholar 

  26. Fabes, E. B., Garofalo, N. andLanconelli, E., Wiener’s criterion for divergence form parabolic operators withC 1-Dini continuous coefficients,Duke Math. J. 59 (1989), 191–232.

    Article  MathSciNet  Google Scholar 

  27. Fabes, E., Jerison, D. andKenig, C., The Wiener test for degenerate elliptic equations,Ann. Inst. Fourier (Grenoble) 32:3 (1982), vi, 151–182.

    MathSciNet  Google Scholar 

  28. Finn, D. L., Behavior of positive solutions to Δ g u=u q+Su with prescribed singularities,Indiana Univ. Math. J. 49 (2000), 177–219.

    MATH  MathSciNet  Google Scholar 

  29. Finn, D. L. andMcOwen, R. C., Singularities and asymptotics for the equation Δ g u=u q+Su, Indiana Univ. Math. J. 42 (1993), 1487–1523.

    Article  MathSciNet  Google Scholar 

  30. Gariepy, R. andZiemer, W. P., A regularity condition at the boundary for solutions of quasilinear elliptic equations,Arch. Rational Mech. Anal. 67 (1977), 25–39.

    Article  MathSciNet  Google Scholar 

  31. Grillot, M. andVéron, L., Boundary trace of the solutions of the prescribed Gaussian curvature equation,Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 527–560.

    MathSciNet  Google Scholar 

  32. Hardy, G. H., Littlewood, J. E. andPólya, G.,Inequalities, Cambridge Univ. Press, Cambridge, 1988.

    Google Scholar 

  33. Havin, V. P. andMaźya, V. G., Nonlinear potential theory,Uspekhi Mat. Nauk 27:6 (1972), 67–138 (Russian). English transl.:Russian Math. Surveys 27:6 (1972), 71–148.

    Google Scholar 

  34. Heinonen, J., Kilpeläinen, T. andMartio, O.,Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993.

    Google Scholar 

  35. Hörmander, L.,Notions of Convexity, Birkhäuser, Boston, Mass., 1994.

    Google Scholar 

  36. Kalton, N. J. andVerbitsky, I. E., Nonlinear equations and weighted norm inequalities,Trans. Amer. Math. Soc. 351 (1999), 3441–3497.

    Article  MathSciNet  Google Scholar 

  37. Keller, J. B., On solutions of δu=f(u), Comm. Pure Appl. Math. 10 (1957), 503–510.

    MATH  MathSciNet  Google Scholar 

  38. Kilpeläinen, T. andMalý, J., The Wiener test and potential estimates for quasilinear elliptic equations,Acta Math. 172 (1994), 137–161.

    MathSciNet  Google Scholar 

  39. Kondratyev, V. A. andNikishkin, V. A., On positive solutions of singular boundary value problems for the equation δu=u k,Russ. J. Math. Phys. 1 (1993), 131–135.

    MathSciNet  Google Scholar 

  40. Kuznetsov, S. E., Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs,Comm. Pure Appl. Math. 51 (1998), 303–340.

    Article  MATH  MathSciNet  Google Scholar 

  41. Kuznetsov, S. E., Removable lateral singularities of semilinear parabolic PDEs and Besov capacities,J. Funct. Anal. 156 (1998), 366–383.

    Article  MATH  MathSciNet  Google Scholar 

  42. Kuznetsov, S. E., Removable singularities forLu=Φ(u) and Orlicz capacities,J. Funct. Anal. 170 (2000), 428–449.

    Article  MATH  MathSciNet  Google Scholar 

  43. Labutin, D. A., Potential estimates for a class of fully nonlinear elliptic equations,Duke Math. J. 111 (2002), 1–49.

    Article  MATH  MathSciNet  Google Scholar 

  44. Labutin, D. A., Thinness for scalar-negative singular Yamabe metrics,In preparation.

  45. Le Gall, J.-F., Nonnegative solutions of δu=u 2 in the unit disk,C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 873–878.

    MATH  Google Scholar 

  46. Le Gall, J.-F., A path-valued Markov process and its connections with partial differential equations, inFirst European Congress of Mathematics (Paris, 1992), Vol. II (Joseph, A., Mignot, F., Murat, F., Prum, B. and Rentschler, R., eds.), pp. 185–212. Birkhäuser, Basel, 1994.

    Google Scholar 

  47. Le Gall, J.-F., A probabilistic Poisson representation for positive solutions of δu=u 2 in a planar domain,Comm. Pure Appl. Math. 50 (1997), 69–103.

    MATH  MathSciNet  Google Scholar 

  48. Le Gall, J.-F., Branching processes, random trees and superprocesses, inProceedings of the International Congress of Mathematicians (Berlin, 1998), Doc. Math., Extra Vol.III, pp. 279–289, 1998.

  49. Le Gall, J.-F.,Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 1999.

    Google Scholar 

  50. Lindqvist, P. andMartio, O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations,Acta Math. 155 (1985), 153–171.

    MathSciNet  Google Scholar 

  51. Littman, W., Stampacchia, G. andWeinberger, H. F., Regular points for elliptic equations with discontinuous coefficients,Ann. Scuola Norm. Sup. Pisa 17 (1963), 43–77.

    MathSciNet  Google Scholar 

  52. Loewner, C. andNirenberg, L., Partial differential equations invariant under conformal or projective transformations, inContributions to Analysis (Ahlfors, L. V., Kra, I., Maskit, B. and Nirenberg, L., eds.), pp. 245–272, Academic Press, New York, 1974.

    Google Scholar 

  53. Malý, J. andZiemer, W. P.,Fine Regularity of Solutions of Elliptic Partial Differential Equations, Amer. Math. Soc., Providence, R. I., 1997.

    Google Scholar 

  54. Marcus, M. andVéron, L., Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations,Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 237–274.

    Article  Google Scholar 

  55. Marcus, M. andVéron, L., The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,Arch. Rational Mech. Anal. 144 (1998), 201–231.

    Article  MathSciNet  Google Scholar 

  56. Marcus, M. andVéron, L., The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case,J. Math. Pures Appl. 77 (1998), 481–524.

    MathSciNet  Google Scholar 

  57. Marcus, M. andVéron, L., Initial trace of positive solutions of some nonlinear parabolic equations,Comm. Partial Differential Equations 24 (1999), 1445–1499.

    MathSciNet  Google Scholar 

  58. Marcus, M. andVéron, L., Removable singularities and boundary traces,J. Math. Pures Appl. 80 (2001), 879–900.

    Article  MathSciNet  Google Scholar 

  59. Maźya, V. G., The continuity at a boundary point of solutions of quasilinear equations,Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25:13 (1970), 42–55. Correction, ibid.27:1 (1972), 160 (Russian). English transl.:Vestnik Leningrad Univ. Math. 3 (1976), 225–242.

    Google Scholar 

  60. Maźya, V. G., Removable singularities of bounded solutions of quasilinear elliptic equations of arbitrary order,Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 26 (1972), 116–130 (Russian). English transl.:J. Soviet Math. 3 (1975), 480–492.

    Google Scholar 

  61. Maźya, V. G.,Sobolev Spaces, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  62. Maźya, V. G., Unsolved problems connected with the Wiener criterion, inThe Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, Mass., 1994) (Jerison, D., Singer, I. M. and Strooch, D. W., eds.), pp. 199–208 Proc. Sympos. Pure Math.60, Amer. Math. Soc., Providence, R. I., 1997.

    Google Scholar 

  63. Maźya, V. G., The Wiener test for higher order elliptic equations,Duke Math. J. 115 (2002), 479–512.

    MathSciNet  Google Scholar 

  64. Maźya, V. G., The Wiener test for higher order elliptic equations, inProceedings of the International Congress of Mathematicians (Beijing, 2002), to appear.

  65. McOwen, R. C., Results and open questions on the singular Yamabe problem, inDynamical Systems and Differential Equations, Vol. II (Springfield, Mo., 1996). Discrete Contin. Dynam. Systems, Added VolumeII (Chen, W. and Hu, S., eds.), pp. 123–132, 1998.

  66. Osserman, R., On the inequality δu≥f(u), Pacific J. Math. 7 (1957), 1641–1647.

    MATH  MathSciNet  Google Scholar 

  67. Perkins, E. A., Polar sets and multiple points for super-Brownian motion,Ann. Probab. 18 (1990), 453–491.

    MATH  MathSciNet  Google Scholar 

  68. Perkins, E. A., Measure-valued branching diffusions and interactions, inProceedings of the International Congress of Mathematicians (Zürich, 1994), Vol.2 (Chatterji, S. D., ed.), pp. 1036–1046, Birkhäuser, Basel, 1995.

    Google Scholar 

  69. Schoen, R. andYau, S.-T., Conformally flat manifolds, Kleinian groups and scalar curvature,Invent. Math. 92 (1988), 47–71.

    Article  MathSciNet  Google Scholar 

  70. Skrypnik, I. V.,Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Amer. Math. Soc., Providence, R. I., 1994.

    Google Scholar 

  71. Trudinger, N. S. andWang, X.-J., Hessian measures I,Topol. Methods Nonlinear Anal. 10 (1997), 225–239.

    MathSciNet  Google Scholar 

  72. Trudinger, N. S. andWang, X.-J., Hessian measures II,Ann. of Math. 150 (1999), 579–604.

    MathSciNet  Google Scholar 

  73. Trudinger, N. S. andWang, X.-J., On the weak continuity of elliptic operators and applications to potential theory,Amer. J. Math. 124 (2002), 369–410.

    MathSciNet  Google Scholar 

  74. Trudinger, N. S. andWang, X.-J., Hessian measures III,J. Funct. Anal. 193 (2002), 1–23.

    Article  MathSciNet  Google Scholar 

  75. Vázquez, J. L. andVéron, L., Singularities of elliptic equations with an exponential nonlinearity,Math. Ann. 269 (1984), 119–135.

    MathSciNet  Google Scholar 

  76. Véron, L.,Singularities of Solutions of Second Order Quasilinear Equations, Longman, Harlow, 1996.

    Google Scholar 

  77. Véron, L., Generalized boundary value problem for nonlinear elliptic equations, inProceedings of the USA-Chile Workhop on Nonlinear Analysis (Vinã del Mar-Valparaiso, 2000) (Manasevich, R. and Rabinowitz, P., eds.),Electron. J. Differential Equations 6, pp. 313–342. Southwest Texas State Univ., San Marcos, Tex., 2001.

    Google Scholar 

  78. Wiener, N., Certain notions in potential theory,J. Math. Phys. 3 (1924), 24–51.

    MATH  Google Scholar 

  79. Wiener, N., The Dirichlet problem,J. Math. Phys. 3 (1924), 127–146.

    MATH  Google Scholar 

  80. Ziemer, W. P.,Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Springer-Verlag, New York, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, 78712-1082, Austin, TX, USA

    Denis A. Labutin

Authors
  1. Denis A. Labutin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Labutin, D.A. Wiener regularity for large solutions of nonlinear equations. Ark. Mat. 41, 307–339 (2003). https://doi.org/10.1007/BF02390818

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02390818

Keywords

Search

Navigation