Convexity of level lines of Martin functions and applications

  • A.-K. Gallagher
  • J. Lebl
  • K. Ramachandran


Let \(\Omega \) be an unbounded domain in \(\mathbb {R}\times \mathbb {R}^{d}.\) A positive harmonic function u on \(\Omega \) that vanishes on the boundary of \(\Omega \) is called a Martin function. In this note, we show that, when \(\Omega \) is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition \(\Omega \) has certain symmetry with respect to the t-axis, and \(\partial \Omega \) is sufficiently flat, then the maximum of any Martin function along a slice \(\Omega \cap (\{t\}\times \mathbb {R}^d)\) is attained at (t, 0).



We thank Alexandre Eremenko for useful discussions and suggestions. We are also greatly indebted to the referee for pointing out an error in an earlier version of Theorem 1.2.

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Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York (1973)Google Scholar
  2. 2.
    Armitage, D.H., Gardiner, S.: Classical Potential Theory. Springer Mongographs in Mathematics. Springer, New York (2001)Google Scholar
  3. 3.
    Caffarelli, L., Spruck, J.: Convexity properties of solutions to some classical variational problems. Commun. Partial Differ. Equ. 7, 1337–1379 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang, S.-Y., Ma, X.-N., Yang, P.: Principal curvature estimates for the convex level sets of semilinear elliptic equations. Discrete Contin. Dyn. Syst. 28(3), 1151–1164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    DeBlassie, D.: The Martin kernel for unbounded domains. Potential Anal. 32(4), 389–404 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gabriel, R.: A result concerning convex level surfaces of 3-dimensional harmonic functions. J. Lond. Math. Soc. 32, 286–294 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhauser, Basel (1994)Google Scholar
  8. 8.
    Korevaar, N.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32, 603–614 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lewis, J.: Capacitary functions in convex rings. Arch. Rat. Mech. Anal. 66, 201–224 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Longinetti, M.: Convexity of the level lines of harmonic functions. Boll. Un. Mat. Ital. A (Italian) 6, 71–75 (1983)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Miyamoto, I.: Harmonic functions in a cylinder which vanish on the boundary. Jpn. J. Math. 22(2), 241–255 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)Google Scholar
  13. 13.
    Ramachandran, K.: Asympotics of positive harmonic functions in certain unbounded domains. Potential Anal. 41(2), 383–405 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Remmert, R.: Classical Topics in Function Theory. Graduate Texts in Mathematics, vol. 122. Springer, New York(1998)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA

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