DOI QR코드

DOI QR Code

ON THE PRESCRIBED MEAN CURVATURE PROBLEM ON THE STANDARD n-DIMENSIONAL BALL

  • Bensouf, Aymen (Department of Mathematics Faculty of sciences of Gafsa Campus Universitaire)
  • 투고 : 2014.11.23
  • 발행 : 2016.03.01

초록

In this paper, we consider the problem of existence of conformal metrics with prescribed mean curvature on the unit ball of ${\mathbb{R}}^n$, $n{\geq}3$. Under the assumption that the order of flatness at critical points of prescribed mean curvature function H(x) is ${\beta}{\in}[1,n-2]$, we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula.

키워드

참고문헌

  1. W. Abdelhedi and H. Chtioui, Prescribing mean curvature on Bn, Internat. J. Math. 21 (2010), no. 9, 1157-1187. https://doi.org/10.1142/S0129167X10006434
  2. W. Abdelhedi, H. Chtioui, and M. Ould Ahmedou, Conformal metrics with prescribed boundary mean curvature on balls, Ann. Global Anal. Geom. 36 (2009), no. 4, 327-362. https://doi.org/10.1007/s10455-009-9157-9
  3. M. A. Al-Ghamdi, H. Chtioui, and K. Sharaf, On a geometric equation involving the Sobolev trace critical exponent, J. Inequal. Appl. 2013 (2013), 405, 25 pp. https://doi.org/10.1186/1029-242X-2013-25
  4. M. A. Al-Ghamdi, H. Chtioui, and K. Sharaf, Topological methods for boundary mean curvature problem on Bn, Adv. Nonlinear Stud. 14 (2014), no. 2, 445-461. https://doi.org/10.1515/ans-2014-0212
  5. A. Bahri, Critical point at infinity in some variational problems, Pitman Res. Notes Math, Ser 182, Longman Sci. Tech. Harlow 1989.
  6. A. Bahri, An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J. 81 (1996), no. 2, 323-466. https://doi.org/10.1215/S0012-7094-96-08116-8
  7. A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimen-sional spheres, J. Funct. Anal. 95 (1991), no. 1, 106-172. https://doi.org/10.1016/0022-1236(91)90026-2
  8. A. Bahri and P. Rabinowitz, Periodic orbits of Hamiltonian systems of three body type, Ann. Inst. H. Poincare Anal. Non Linaire 8 (1991), 561-649. https://doi.org/10.1016/S0294-1449(16)30252-9
  9. M. Ben Ayed, Y. Chen, H. Chtioui, and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J. 84 (1996), no. 3, 633-677. https://doi.org/10.1215/S0012-7094-96-08420-3
  10. R. Ben Mahmoud and H. Chtioui, Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete Contin. Dyn. Syst. 32 (2012), no. 5, 1857-1879. https://doi.org/10.3934/dcds.2012.32.1857
  11. A. Bensouf and H. Chtioui, Conformal metrics with prescribed Q-curvature on $S^n$, Calc. Var. Partial Differential Equations 41 (2011), no. 3-4, 455-481. https://doi.org/10.1007/s00526-010-0372-9
  12. J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ Press, 1965.
  13. Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383-417. https://doi.org/10.1215/S0012-7094-95-08016-8

피인용 문헌

  1. On the boundary mean curvature equation on B n $\mathbb{B}^{n}$ vol.2016, pp.1, 2016, https://doi.org/10.1186/s13661-016-0727-z