The differentiability of solutions for elliptic equations which degenerate on part of the boundary of a convex domain

  • Jia-xin Song
  • Yi Cao


In this paper, we study the differentiability of solutions on the boundary for equations of type \({L_\lambda }u = \frac{{{\partial ^2}u}}{{\partial {x^2}}} + {\left| x \right|^{2\lambda }}\frac{{{\partial ^2}u}}{{\partial {y^2}}} = f\left( {x,y} \right)\) , where λ is an arbitrary positive number. By introducing a proper metric that is related to the elliptic operator Lλ, we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied. The main diffculty is the construction of new barrier functions in this article.


elliptic equations convex domain differentiability 

MR Subject Classification

35J70 35H20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors wish to express our sincere thanks to the referees for their careful reading and helpful comments


  1. [1]
    SX Chen. The fundamental solution of the Keldysh type operator, Sci China Ser A, 2009, 52(9): 1829–1843.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P Daskalopoulos, R Hamilton. Regularity of the free boundary for the porous medium equation, J Amer Math Soc, 1998, 11(4): 899–965.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D Gilbarg, NS Trudinger. Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224.Google Scholar
  4. [4]
    L Hömander. Hypoelliptic second order differential equations, Acta Math, 1967, 119: 147–171.MathSciNetCrossRefGoogle Scholar
  5. [5]
    T Hristov, N Popivanov, M Schneider. Generalized solutions to Protter problems for 3-D Keldysh type equations, AIP Conf Proc, 2014, 1637(1): 422–430.CrossRefGoogle Scholar
  6. [6]
    T Hristov. Singular solutions to Protter problem for Keldysh type equations, AIP Conf Proc, 2014, 1631(1): 255–262.CrossRefGoogle Scholar
  7. [7]
    MV Keldysh. On certain cases of degeneration of equation of elliptic type on the boundary of a domain, Dokl Akad Nauk SSSR, 1951, 77: 181–183.Google Scholar
  8. [8]
    DS Li, LH Wang. Boundary differentiability of solutions of elliptic equations on convex domains, Manuscripta Math, 2006, 121(7): 137–156.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    DS Li, LH Wang. Elliptic equations on convex domains with nonhomogeneous Dirichlet boundary conditions, J Differential Equations, 2009, 246: 1723–1743.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    QZ Song, LH Wang. Hölder estimates for elliptic equations degenerate on part of the boundary of a domain, Manuscrupta Math, 2012, 139(1-2): 179–200.CrossRefzbMATHGoogle Scholar
  11. [11]
    LH Wang. Hölder estimates for subelliptic operators, J Funct Anal, 2003, 199(1): 228–242.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    LH Wang. On the regularity theory of fully nonlinear parabolic equations, Bull Amer Math Soc, 1990, 22(1): 107–114.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

Personalised recommendations