Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points

Jan Malý

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 1, page 23-42
  • ISSN: 0010-2628

Abstract

top
Let u be a weak solution of a quasilinear elliptic equation of the growth p with a measure right hand term μ . We estimate u ( z ) at an interior point z of the domain Ω , or an irregular boundary point z ∈ ∂ Ω , in terms of a norm of u , a nonlinear potential of μ and the Wiener integral of 𝐑 n ∖ Ω . This quantifies the result on necessity of the Wiener criterion.

How to cite

top

Malý, Jan. "Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points." Commentationes Mathematicae Universitatis Carolinae 37.1 (1996): 23-42. <http://eudml.org/doc/247923>.

@article{Malý1996,
abstract = {Let $u$ be a weak solution of a quasilinear elliptic equation of the growth $p$ with a measure right hand term $\mu $. We estimate $u(z)$ at an interior point $z$ of the domain $\Omega $, or an irregular boundary point $z\in \partial \Omega $, in terms of a norm of $u$, a nonlinear potential of $\mu $ and the Wiener integral of $\mathbf \{R\}^n\setminus \Omega $. This quantifies the result on necessity of the Wiener criterion.},
author = {Malý, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {elliptic equations; Wiener criterion; nonlinear potentials; measure data; measure data; Wiener integral; Wiener criterion},
language = {eng},
number = {1},
pages = {23-42},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points},
url = {http://eudml.org/doc/247923},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Malý, Jan
TI - Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 1
SP - 23
EP - 42
AB - Let $u$ be a weak solution of a quasilinear elliptic equation of the growth $p$ with a measure right hand term $\mu $. We estimate $u(z)$ at an interior point $z$ of the domain $\Omega $, or an irregular boundary point $z\in \partial \Omega $, in terms of a norm of $u$, a nonlinear potential of $\mu $ and the Wiener integral of $\mathbf {R}^n\setminus \Omega $. This quantifies the result on necessity of the Wiener criterion.
LA - eng
KW - elliptic equations; Wiener criterion; nonlinear potentials; measure data; measure data; Wiener integral; Wiener criterion
UR - http://eudml.org/doc/247923
ER -

References

top
  1. Adams D.R., L p potential theory techniques and nonlinear PDE, In: Potential Theory (Ed. M. Kishi) Walter de Gruyter & Co Berlin (1992), 1-15. (1992) Zbl0760.22013MR1167217
  2. Adams D.R., Hedberg L.I., Function Spaces and Potential Theory, Springer Verlag Berlin (1995). (1995) Zbl0834.46021MR1411441
  3. Adams D.R., Meyers N.G., Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972), 169-197. (1972) Zbl0244.31012MR0316724
  4. Brelot M., On Topologies and Boundaries in Potential Theory, Lecture Notes in Math. 175, Springer ({1971}). ({1971}) Zbl0222.31014MR0281940
  5. Federer H., Ziemer W.P., The Lebesgue set of a function whose partial derivatives are p -th power summable, Indiana Univ. Math. J. 22 (1972), 139-158. (1972) MR0435361
  6. Frehse J., Capacity methods in the theory of partial differential equations, Jahresber. Deutsch. Math. Verein. 84 (1982), 1-44. (1982) Zbl0486.35002MR0644068
  7. Fuglede B., The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier Grenoble 21.1 (1971), 123-169. (1971) Zbl0197.19401MR0283158
  8. Gariepy R., Ziemer W.P., A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rat. Mech. Anal. 67 (1977), 25-39. (1977) Zbl0389.35023MR0492836
  9. Hedberg L.I., Nonlinear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299-319. (1972) MR0328088
  10. Hedberg L.I., Wolff Th.H., Thin sets in nonlinear potential theory, Ann. Inst. Fourier 33.4 (1983), 161-187. (1983) Zbl0508.31008MR0727526
  11. Heinonen J., Kilpeläinen T., On the Wiener criterion and quasilinear obstacle problems, Trans. Amer. Math. Soc. 310 (1988), 239-255. (1988) MR0965751
  12. Heinonen J., Kilpeläinen T., Martio O., Fine topology and quasilinear elliptic equations, Ann. Inst. Fourier 39.2 (1989), 293-318. (1989) MR1017281
  13. Heinonen J., Kilpeläinen T., Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford (1993). (1993) MR1207810
  14. Kilpeläinen T., Malý J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa. Cl. Science, Ser. IV 19 (1992), 591-613. (1992) MR1205885
  15. Kilpeläinen T., Malý J., Supersolutions to degenerate elliptic equations on quasi open sets, Comm. Partial Differential Equations 17 (1992), 371-405. (1992) MR1163430
  16. Kilpeläinen T., Malý J., The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161. (1994) MR1264000
  17. Lieberman G.M., Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations with right hand side a measure, Comm. Partial Differential Equations 18 (1993), 1991-2112. (1993) MR1233190
  18. Lindqvist P., Martio O., Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171. (1985) Zbl0607.35042MR0806413
  19. Littman W., Stampacchia G., Weinberger H.F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa. Serie III 17 (1963), 43-77. (1963) Zbl0116.30302MR0161019
  20. Malý J., Nonlinear potentials and quasilinear PDE's, Proceedings of the International Conference on Potential Theory, Kouty, 1994, to appear. Zbl0857.35046MR1404703
  21. Maz'ya V.G., On the continuity at a boundary point of solutions of quasi-linear elliptic equations (Russian), Vestnik Leningrad. Univ. 25 42-55 English translation Vestnik Leningrad. Univ. Math. 3 (1976), 225-242. (1976) MR0274948
  22. Maz'ya V.G., Khavin V.P., Nonlinear potential theory (Russian), Uspekhi Mat. Nauk 27.6 (1972), 67-138 English translation Russian Math. Surveys 27 (1972), 71-148. (1972) 
  23. Malý J., Ziemer W.P., Fine Regularity of Solutions of Elliptic Equations, book in preparation. 
  24. Meyers N.G., Continuity properties of potentials, Duke Math. J. 42 (1975), 157-166. (1975) Zbl0334.31004MR0367235
  25. Rakotoson J.M., Ziemer W.P., Local behavior of solutions of quasilinear elliptic equations with general structure, Trans. Amer. Math. Soc. 319 (1990), 747-764. (1990) Zbl0708.35023MR0998128
  26. Skrypnik I.V., Nonlinear Elliptic Boundary Value Problems, Teubner Verlag, Leipzig (1986). (1986) Zbl0617.35001MR0915342
  27. Trudinger N.S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. (1967) Zbl0153.42703MR0226198
  28. Wiener N., Certain notions in potential theory, J. Math. Phys. 3 (1924), 24-5 Reprinted in: Norbert Wiener: Collected works. Vol. 1 (1976), MIT Press, pp. 364-391. (1924) MR0532698

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.