Duke Mathematical Journal

Potential estimates for a class of fully nonlinear elliptic equations

Denis A. Labutin

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Abstract

We study the pointwise properties of $k$-subharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations $F_k[u]=0$, where $F_k[u]$ is the elementary symmetric function of order $k,1\leq k\leq n$, of the eigenvalues of $[D\sp 2u]$, $F_1[u]=\Delta u,F_n[u]=\det D^2u$. Thus $1$-subharmonic functions are subharmonic in the classical sense; $n$-subharmonic functions are convex. We use a special capacity to investigate the typical questions of potential theory: local behaviour, removability of singularities, and polar, negligible, and thin sets, and we obtain estimates for the capacity in terms of the Hausdorff measure. We also prove the Wiener test for the regularity of a boundary point for the Dirichlet problem for the fully nonlinear equation $F_k[u]=0$. The crucial tool in the proofs of these results is the Radon measure $F_k[u]$ introduced recently by N. Trudinger and X.-J. Wang for any $k$-subharmonic $u$. We use ideas from the potential theories both for the complex Monge-Ampère and for the $p$-Laplace equations.

Article information

Source
Duke Math. J., Volume 111, Number 1 (2002), 1-49.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575006

Digital Object Identifier
doi:10.1215/S0012-7094-02-11111-9

Mathematical Reviews number (MathSciNet)
MR1876440

Zentralblatt MATH identifier
1100.35036

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 31B15: Potentials and capacities, extremal length 31C45: Other generalizations (nonlinear potential theory, etc.)

Citation

Labutin, Denis A. Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111 (2002), no. 1, 1--49. doi:10.1215/S0012-7094-02-11111-9. https://projecteuclid.org/euclid.dmj/1087575006


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References

  • 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.
  • --. --. --. --., review of Fine Regularity of Solutions of Elliptic Partial Differential Equations by J. Malý and W. P. Ziemer, Bull. London Math. Soc. 31 (1999), 248--250.
  • D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
  • E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1--40.
  • L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, Amer. Math. Soc., Providence, 1995.
  • 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.
  • H. Cartan, Théorie du potentiel newtonien: énergie, capacité, suites de potentiels, Bull. Soc. Math. France 73 (1945), 74--106.
  • G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953), 131--295.
  • 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.
  • 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.
  • B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209--243.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2d ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983.
  • B. Guan, The Dirichlet problem for a class of fully nonlinear elliptic equations, Comm. Partial Differential Equations 19 (1994), 399--416.
  • B. Guan and J. Spruck, Boundary-value problems on (S^n) for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), 601--624.
  • W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Vol. I, London Math. Soc. Monogr. 9, Academic Press, London, 1976.
  • L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161--187.
  • J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monogr., Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
  • L. Hörmander, Notions of Convexity, Progr. Math. 127, Birkhäuser, Boston, 1994.
  • N. M. Ivochkina, 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.; English translation in Math. USSR-Sb. 50, no. 1 (1985), 259--268.
  • --. --. --. --., 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; English translation in Math. USSR-Sb. 56, no. 2 (1987), 403--415.
  • T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137--161.
  • M. Klimek, Pluripotential Theory, London Math. Soc. Monogr. (N.S.) 6, Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.
  • N. V. Krylov, Lectures on fully nonlinear elliptic equations, Rudolph Lipschitz Lectures, Univ. of Bonn, Germany, 1993.
  • D. A. Labutin, Pluripolarity of sets with small Hausdorff measure, Manuscripta Math. 102 (2000), 163--167.
  • --------, Isolated singularities of solutions of fully nonlinear elliptic equations, to appear in J. Differential Equations.
  • N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, New York, 1972.
  • P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153--171.
  • J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys Monogr. 51, Amer. Math. Soc., Providence, 1997.
  • 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.; English translation in Vestnik Leningrad. Univ. Math. 3 (1976), 225--242.
  • --. --. --. --., ``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.
  • V. G. Maz'ya and V. P. Havin, A nonlinear potential theory (in Russian), Uspekhi Mat. Nauk 27, no. 6 (1972), 67--138.; English translation in Russian Math. Surveys 27, no. 6 (1972), 71--148.
  • J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965), 219--240.
  • 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.
  • --. --. --. --., The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), 153--179.
  • --. --. --. --., Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 411--425.
  • --. --. --. --., On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151--164.
  • --. --. --. --., On new isoperimetric inequalities and symmetrization, J. Reine Angew. Math. 488 (1997), 203--220.
  • --. --. --. --., Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), 1251--1261.
  • N. S. Trudinger and X.-J. Wang, Hessian measures, I, Topol. Methods Nonlinear Anal. 10 (1997), 225--239.
  • --. --. --. --., Hessian measures, II, Ann. of Math. (2) 150 (1999), 579--604.
  • --------, Hessian measures, III, preprint, 2000, Australian National University mathematics research report no. MRR00-016, http://www.math.anu.edu.au/research.reports/00mrr.html
  • 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.
  • J. Väisälä, Capacity and measure, Michigan Math. J. 22 (1975), 1--3.
  • X. J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994), 25--54.