Potential estimates for a class of fully nonlinear elliptic equations

Duke Mathematical Journal

Volumn 111, Issue 1, 2002, Pages 1-49

  Labutin, Denis A ^a,b  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037082247     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-02-11111-9     Document Type: Article

References (48)

1. D. R. ADAMS, “Potential and capacity before and after Wiener” in Proceedings of the Norbert Wiener Centenary Congress (East Lansing, Mich., 1994), Proc. Sympos. Appl. Math. 52, Amer. Math. Soc., Providence, 1997, 63-83.
2. D. R. ADAMS, review of Fine Regularity of Solutions of Elliptic Partial Differential Equations by J. Maly and W. P. Ziemer, Bull. London Math. Soc. 31 (1999), 248-250.
3. D. R. ADAMS and L. I. HEDBERG, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
4. E. BEDFORD and B. A. TAYLOR, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40.
5. L. A. CAFFARELLI and X. CABRE, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, Amer. Math. Soc., Providence, 1995.
6. L. CAFFARELLI, L. NIRENBERG, and J. SPRUCK, The Dirichlet problem for nonlinear second-order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301.
7. H. CARTAN, Théorie dupotentielnewtonien: énergie, capacité, suites depotentiels, Bull. Soc. Math. France 73 (1945), 74-106.
8. G. CHOQUET, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953), 131-295.
9. M. G. CRANDALL, “Viscosity solutions: A primer” in Viscosity Solutions and Applications (Montecatini Terme, Italy, 1995), Lecture Notes in Math. 1660, Springer, Berlin, 1997, 1-43.
10. M. G. CRANDALL, H. ISHII, and P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67.
11. B. GIDAS, W. M. NI, and L. NIRENBERG, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.
12. D. GILBARG and N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2d ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983.
13. B. GUAN, The Dirichlet problem for a class of fully nonlinear elliptic equations, Comm. Partial Differential Equations 19 (1994), 399-416.
14. n for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), 601-624.
15. W. K. HAYMAN and P. B. KENNEDY, Subharmonic Functions, Vol. I, London Math. Soc. Monogr. 9, Academic Press, London, 1976.
16. L. I. HEDBERG and TH. H. WOLFF, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161-187.
17. J. HEINONEN, T. KILPELÀINEN, and O. MARTIO, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monogr., Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
18. L. HORMANDER, Notions of Convexity, Progr. Math. 127, Birkhauser, Boston, 1994.
19. N. M. IVOCHKINÀ, Description of cones of stability generated by differential operators of Monge -Ampère type (in Russian), Mat. Sb. (N.S.) 122 (164), no. 2 (1983), 265-275.
20. N. M. IVOCHKINÀ, Description of cones of stability generated by differential operators of Monge -Ampère type (in Russian), English translation in Math. USSR-Sb. 50, no. 1 (1985), 259-268.
21. N. M. IVOCHKINÀ, Solution of the Dirichlet problem for certain equations of Monge-Ampère type (in Russian), Mat. Sb. (N.S.) 128 (170), no. 3 (1985), 403-415, 447.
22. N. M. IVOCHKINÀ, Solution of the Dirichlet problem for certain equations of Monge-Ampère type (in Russian), English translation in Math. USSR-Sb. 56, no. 2 (1987), 403-415.
23. T. KILPELÀINEN andJ. MALY, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137-161.
24. M. KLIMEK, Pluripotential Theory, London Math. Soc. Monogr. (N.S.) 6, Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.
25. N. V. KRYLOV, Lectures on fully nonlinear elliptic equations, Rudolph Lipschitz Lectures, Univ. of Bonn, Germany, 1993.
26. D. A. LÀBUTIN, Pluripolarity of sets with small Hausdorff measure, Manuscripta Math. 102 (2000), 163-167.
27. D. A. LÀBUTIN, Isolated singularities of solutions of fully nonlinear elliptic equations, to appear in J. Differential Equations.
28. N. S. LANDKOF, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, New York, 1972.
29. P. LINDQVIST and O. MARTIO, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171.
30. J. MALY and W. P. ZIEMER, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys Monogr. 51, Amer. Math. Soc., Providence, 1997.
31. V. G. MAZ’YA, The continuity at a boundary point of the solutions of quasi-linear elliptic equations (in Russian), Vestnik Leningrad. Univ. 25, no. 13 (1970). 42-55.
32. V. G. MAZ’YA, The continuity at a boundary point of the solutions of quasi-linear elliptic equations (in Russian), English translation in Vestnik Leningrad. Univ. Math. 3 (1976), 225-242.
33. V. G. MAZ’YA, “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.
34. V. G. MAZ’YA and V. P. HAVIN, A nonlinear potential theory (in Russian), Uspekhi Mat. Nauk 27, no. 6 (1972), 67-138.
35. V. G. MAZ’YA and V. P. HAVIN, A nonlinear potential theory (in Russian), English translation in Russian Math. Surveys 27, no. 6 (1972), 71-148.
36. J. SERRIN, Isolated singularities of solutions of quasilinear equations, Acta Math. 113 (1965), 219-240.
37. N. S. TRUDINGER, “A priori bounds for graphs with prescribed curvature” in Analysis, et cetera: Research Papers Published in Honor of Jürgen Moser's 60th Birthday, Academic Press, Boston, 1990, 667-676.
38. N. S. TRUDINGER, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), 153-179.
39. N. S. TRUDINGER, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincare Anal. Non Lineaire 11 (1994), 411-425.
40. N. S. TRUDINGER, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164.
41. N. S. TRUDINGER, On new isoperimetric inequalities and symmetrization, J. Reine Angew. Math. 488 (1997), 203-220.
42. N. S. TRUDINGER, Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), 1251-1261.
43. N. S. TRUDINGER and X.-J. WANG, Hessian measures, I, Topol. Methods Nonlinear Anal. 10 (1997), 225-239.
44. N. S. TRUDINGER and X.-J. WANG, Hessian measures, II, Ann. of Math. (2) 150 (1999), 579-604.
45. N. S. TRUDINGER and X.-J. WANG, Hessian measures, III, preprint, 2000, Australian National University mathematics research report no. MRR00-016, http://www.math.anu.edu.au/research.reports/00mrr.html
46. J. I. E. URBAS, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), 355-382.
47. J. VÀISÀLÀ, Capacity and measure, Michigan Math. J. 22 (1975), 1-3.
48. X. J. WANG, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994), 25-54.