The Wiener test for higher order elliptic equations

Duke Mathematical Journal

Volumn 115, Issue 3, 2002, Pages 479-512

  Maz'ya, Vladimir ^a  

^a LINKÖPING UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036925940     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-02-11533-6     Document Type: Article

References (32)

1. [AH] D.R. Adams and A. Heard, The necessity of the Wiener test for some semi-linear elliptic equations, Indiana Univ. Math. J. 41 (1992), 109-124. MR 94c:35080.
2. [AHe] D.R. Adams and L.I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. MR 97j:46024.
3. [ADN] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 23:A2610.
4. [DMM] G. Dal Maso and U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986), 345-387. MR 87m:35021.
5. p regularity of self-adjoint elliptic operators, J. Differential Equations 135 (1997), 83-102. MR 98i:47044.
6. [E] S. Eilertsen, On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian, Ark. Mat. 38 (2000), 53-75. MR 2001f:31009.
7. [FJK] E. Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), 151-182. MR 84g:35067.
8. [H] L.I. Hedberg, Approximation by harmonic functions, and stability of the Dirichlet problem, Exposition. Math. 11 (1993), 193-259. MR 94k:31001.
9. [J] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955. MR 17:746d.
10. [KM] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161. MR 95a:35050.
11. [KKM] V.A. Kozlov, V.A. Kondrat'ev, and V.G. Maz'ya, On sign variability and the absence of "strong" zeros of solutions of elliptic equations, Math. USSR-Izv. 34 (1990), 337-353. MR 90i:35021.
12. [KMR] V.A. Kozlov, V.G. Maz'ya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Math. Surveys Monogr. 85, Amer. Math. Soc., Providence, 2001. MR 2001i:35069.
13. [L] N.S. Landkof, Foundations of Modern Potential Theory (in Russian), "Nauka" Moscow, 1966, MR 35:5644; English translation in Grundlehren Math. Wiss. 180, Springer, New York, 1972. MR 50:2520.
14. [L] N.S. Landkof, Foundations of Modern Potential Theory (in Russian), "Nauka" Moscow, 1966, MR 35:5644; English translation in Grundlehren Math. Wiss. 180, Springer, New York, 1972. MR 50:2520.
15. [La] D.A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002), 1-49. CMP 1 876 440.
16. [LM] P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171. MR 87g:35074.
17. [LSW] W. Littman, G. Stampacchia, and H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43-77. MR 28:4228.
18. [MZ] J. Malý and W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys Monogr. 51, Amer. Math. Soc., Providence, 1997. MR 98h:35080.
19. [M1] V.G. Maz'ya [V.G. Maz'ja], The Dirichlet problem for elliptic equations of arbitrary order in unbounded domains (in Russian), Dokl. Akad. Nauk SSSR 150 (1963), 1221-1224, MR 27:5026; English translation in Soviet Math. Dokl. 4 (1963), 860-863.
20. [M1] V.G. Maz'ya [V.G. Maz'ja], The Dirichlet problem for elliptic equations of arbitrary order in unbounded domains (in Russian), Dokl. Akad. Nauk SSSR 150 (1963), 1221-1224, MR 27:5026; English translation in Soviet Math. Dokl. 4 (1963), 860-863.
21. [M2] _, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, 1985. MR 87g:46056.
22. [M3] _, "Unsolved problems connected with the Wiener criterion" in The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, Mass., 1994), Proc. Sympos. Pure Math. 60, Amer. Math. Soc., Providence, 1997, 199-208. MR 98e:35040.
23. [M4] _, "On Wiener's type regularity of a boundary point for higher order elliptic equations" in Nonlinear Analysis, Function Spaces and Applications (Prague, 1998), Vol. 6, Acad. Sci. Czech Repub., Prague, 1999, 119-155. MR 2001m:35129.
24. [MN] V.G. Maz'ya and S.A. Nazarov, The apex of a cone can be irregular in Wiener's sense for a fourth-order elliptic equation (in Russian), Mat. Zametki 39 (1986), 24-28; English translation in Math. Notes 39 no. 1, (1986), 14-16. MR 87h:35088.
25. [MN] V.G. Maz'ya and S.A. Nazarov, The apex of a cone can be irregular in Wiener's sense for a fourth-order elliptic equation (in Russian), Mat. Zametki 39 (1986), 24-28; English translation in Math. Notes 39 no. 1, (1986), 14-16. MR 87h:35088.
26. [MNP] V. Maz'ya [W.G. Mazja], S. Nazarov [s.A. Nasarow], and B. Plamenevskij [B.A. Plamenewski], Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, I: Störungen isolierter Randsingularitäten, Math. Lehrbucher Monogr. I. Abt. Math. Lehrbucher 82, Akademie, Berlin, 1991, MR 92g:35059; English translation in Oper. Theory Adv. Appl. 111, Birkhäuser, Basel, 2000. MR 2001e:35044a.
27. [MNP] V. Maz'ya [W.G. Mazja], S. Nazarov [S.A. Nasarow], and B. Plamenevskij [B.A. Plamenewski], Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, I: Störungen isolierter Randsingularitäten, Math. Lehrbucher Monogr. I. Abt. Math. Lehrbucher 82, Akademie, Berlin, 1991, MR 92g:35059; English translation in Oper. Theory Adv. Appl. 111, Birkhäuser, Basel, 2000. MR 2001e:35044a.
28. p-estimates of solutions of elliptic boundary value problems in domains with ribs (in Russian), Trudy Moskov. Mat. Obshch. 37 (1978), 49-93, MR 81b:35027; English translation in Trans. Moscow Math. Soc. 37 (1980), 49-97.
29. [MP2] _, On the maximum principle for the biharmonic equation in a domain with conical points (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1981, no. 2, 52-59, MR 84b:35037.
30. [TW] N.S. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 369-410. CMP 1 890 997.
31. [W] N. Wiener, The Dirichlet problem, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 127-146; reprinted in N. Wiener, Collected Works with Commentaries, Vol. 1, ed. P. Masani, Mathematicians of Our Time 10, MIT Press, Cambridge, Mass., 1976, 394-413. MR 58:27161.
32. [W] N. Wiener, The Dirichlet problem, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 127-146; reprinted in N. Wiener, Collected Works with Commentaries, Vol. 1, ed. P. Masani, Mathematicians of Our Time 10, MIT Press, Cambridge, Mass., 1976, 394-413. MR 58:27161.