On the Existence of \(L^p\)-Solution of Generalized Euler–Poisson–Darboux Equation in the Upper Half Space

  • Kangqun ZhangEmail author


We focus on the existence of solution of generalized Euler–Poisson–Darboux equation, which is elliptic in \(\mathbb R^{n+1}_+\) and has a singular coefficient on its boundary. Based on Mikhlin’s multiplier theorem and Hardy inequalities, the well-posedness of its Dirichlet problem in the upper half space is established.


Generalized Euler–Poisson–Darboux equation Elliptic type \(L^p\)-solution 

Mathematics Subject Classification

35Q05 35J75 



We sincerely thank the anonymous referees for their comments and useful suggestions. This work was partially supported by NNSF of China (11326152), NSF of Jiangsu Province of China (BK20130736), NSF of the Jiangsu Higher Education Institutions of China (18KJB110013) and NSF of Nanjing Institute of Technology (CKJB201709).


  1. Algazin, O.D.: Construction by similarity method of the fundamental solution of the Dirichlet problem for Keldysh type equation in the half-space (2016). arXiv preprint. arXiv:1612.00205
  2. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer Science & Business Media, Berlin (2012)zbMATHGoogle Scholar
  3. Bitsadze, A.V.: Some Classes of Partial Differential Equations. CRC Press, Boca Raton (1988)zbMATHGoogle Scholar
  4. Darboux, G.: Leons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, vol. 2. Gauthier-Villars, Paris (1915)zbMATHGoogle Scholar
  5. Diaz J.B., Weinstein A.: On the fundamental solutions of a singular Beltrami operator. In: Studies in Mathematics and Mechanics Presented to Richard von Mises, pp. 97–102 (2013)Google Scholar
  6. Euler, L.: Institutiones calculi integralis. Academia Imperialis Scientiarum, Saint Petersburg (1792)Google Scholar
  7. Fornaro, S., Metafune, G., Pallara, D., et al.: Degenerate operators of Tricomi type in \(L^p\) spaces and in spaces of continuous functions. J. Differ. Equ. 252(2), 1182–1212 (2012)CrossRefzbMATHGoogle Scholar
  8. Fornaro, S., Metafune, G., Pallara, D., et al.: Second order elliptic operators in \(L^2\) with first order degeneration at the boundary and outward pointing drift. Commun. Pure Appl. Anal. 14, 407419 (2015)zbMATHGoogle Scholar
  9. Germain, P., Trudinger, N.S.: Solutions élémentaires de certaines équations aux dérivées partielles du type mixte. Bull. Soc. Math. Fr. 81, 145–174 (1953)CrossRefzbMATHGoogle Scholar
  10. Han, Q., Hong, J., Huang, G.: Compactness of Alexandrov–Nirenberg surfaces. Commun. Pure Appl. Math. 70(9), 1706–1753 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Huang, G., Li, C.: A Liouville theorem for high order degenerate elliptic equations. J. Differ. Equ. 258(4), 1229–1251 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Keldysh, M.V.: On some cases of degenerate elliptic equations on the boundary of a domain. Dokl. Acad. Nauk USSR 77, 181–183 (1951)Google Scholar
  13. Kim, J.U.: An \(L^p\) a priori estimate for the Tricomi equation in the upper half space. Trans. Am. Math. Soc. 351(11), 4611–4628 (1999)CrossRefzbMATHGoogle Scholar
  14. Konopelchenko, B.G., Ortenzi, G.: Elliptic Euler–Poisson–Darboux equation, critical points and integrable systems. J. Phys. A Math. Theor. 46(48), 485204 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kuang, J.C.: Applied Inequalities. Shandong Science and Technology Press, Jinan (2004)Google Scholar
  16. McLachlan, N.W.: Bessel Functions for Engineers. Oxford Clarendon Press, Oxford (1955)zbMATHGoogle Scholar
  17. Prüss, J.: On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift. Arch. Math. 108, 301–311 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Smirnov, M.M.: Equations of mixed type. Vyssh. Shkola, Moscow (1985)Google Scholar
  19. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  20. Xu, N., Yin, H.: The weighted \(W^{2, p}\) estimate on the solution of the Gellerstedt equation in the upper half space. J. Math. Anal. Appl. 332(2), 1148–1164 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsNanjing Institute of TechnologyNanjingChina

Personalised recommendations