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On a class of semipositone problems with singular Trudinger-Moser nonlinearities
Compactness results for linearly perturbed Yamabe problem on manifolds with boundary
Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56126 Pisa, Italy |
Let $ (M,g) $ a compact Riemannian $ n $-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of $ g $ there are scalar-flat metrics that have $ \partial M $ as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term $ h_{g} $ with a negative smooth function $ \alpha, $ the set of solutions of Yamabe problem is still a compact set.
References:
| [1] |
S. de M. Almaraz,
A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.
doi: 10.1007/s00526-010-0365-8. |
| [2] |
S. de M. Almaraz,
Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.
doi: 10.1016/j.jde.2011.04.013. |
| [3] |
S. de M. Almaraz,
An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.
doi: 10.2140/pjm.2010.248.1. |
| [4] |
S. Almaraz, O. S. de Queiroz and S. Wang,
A compactness theorem for scalar-flat metrics on 3-manifolds with boundary, J. Funct. Anal., 277 (2019), 2092-2116.
doi: 10.1016/j.jfa.2019.01.001. |
| [5] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
| [6] |
S. S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302, (2010). |
| [7] |
O. Druet,
Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 2004 (2004), 1143-1191.
doi: 10.1155/S1073792804133278. |
| [8] |
O. Druet and E. Hebey,
Blow-up examples for second order elliptic PDEs of critical sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.
doi: 10.1090/S0002-9947-04-03681-5. |
| [9] |
J. F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math., 136, (1992), 1–50.
doi: 10.2307/2946545. |
| [10] |
V. Felli and M. Ould Ahmedou,
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.
doi: 10.1007/s00209-002-0486-7. |
| [11] |
M. Ghimenti and A. M. Micheletti, Compactness for conformal scalar-flat metrics on umbilic boundary manifolds, Nonlinear Anal, 200 (2020), 30 pp
doi: 10.1016/j.na.2020.111992. |
| [12] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary, J. Differential Equations, 267 (2019), 587-618.
doi: 10.1016/j.jde.2019.01.023. |
| [13] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Linear perturbation of the Yamabe problem on manifolds with boundary, J. Geom. Anal., 28 (2018), 1315-1340.
doi: 10.1007/s12220-017-9864-6. |
| [14] |
G. Giraud,
Sur le probléme de Dirichlet généralisé, Ann. Sci. École Norm. Sup., 46 (1929), 131-145.
|
| [15] |
Z.-C. Han and Y. Li,
The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math, J., 99 (1999), 489-542.
doi: 10.1215/S0012-7094-99-09916-7. |
| [16] |
S. Kim, M. Musso and J. Wei, Compactness of scalar-flat conformal metrics on low-dimensional manifolds manifolds with constant mean curvature on boundary, preprint, arXiv: 1906.01317. |
| [17] |
Y. Li and M. Zhu,
Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
| [18] |
F. C. Marques,
Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.
doi: 10.1512/iumj.2005.54.2590. |
| [19] |
F. C. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., 71 (2005), 315-346.
doi: 10.4310/jdg/1143651772. |
| [20] |
F. C. Marques,
Compactness and non compactness for Yamabe-type problems, Progress in Nonlinear Differential Equation and their Applications, 86 (2015), 121-131.
doi: 10.1007/978-3-319-19902-3_9. |
| [21] |
M. Mayer and C. B. Ndiaye,
Barycenter technique and the Riemann mapping problem of Cherrier-Escobar, J. Differential Geom., 107 (2017), 519-560.
doi: 10.4310/jdg/1508551224. |
| [22] |
R. Schoen and D. Zhang,
Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differ. Equ., 4 (1996), 1-25.
doi: 10.1007/BF01322307. |
show all references
References:
| [1] |
S. de M. Almaraz,
A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.
doi: 10.1007/s00526-010-0365-8. |
| [2] |
S. de M. Almaraz,
Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.
doi: 10.1016/j.jde.2011.04.013. |
| [3] |
S. de M. Almaraz,
An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.
doi: 10.2140/pjm.2010.248.1. |
| [4] |
S. Almaraz, O. S. de Queiroz and S. Wang,
A compactness theorem for scalar-flat metrics on 3-manifolds with boundary, J. Funct. Anal., 277 (2019), 2092-2116.
doi: 10.1016/j.jfa.2019.01.001. |
| [5] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
| [6] |
S. S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302, (2010). |
| [7] |
O. Druet,
Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 2004 (2004), 1143-1191.
doi: 10.1155/S1073792804133278. |
| [8] |
O. Druet and E. Hebey,
Blow-up examples for second order elliptic PDEs of critical sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.
doi: 10.1090/S0002-9947-04-03681-5. |
| [9] |
J. F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math., 136, (1992), 1–50.
doi: 10.2307/2946545. |
| [10] |
V. Felli and M. Ould Ahmedou,
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.
doi: 10.1007/s00209-002-0486-7. |
| [11] |
M. Ghimenti and A. M. Micheletti, Compactness for conformal scalar-flat metrics on umbilic boundary manifolds, Nonlinear Anal, 200 (2020), 30 pp
doi: 10.1016/j.na.2020.111992. |
| [12] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary, J. Differential Equations, 267 (2019), 587-618.
doi: 10.1016/j.jde.2019.01.023. |
| [13] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Linear perturbation of the Yamabe problem on manifolds with boundary, J. Geom. Anal., 28 (2018), 1315-1340.
doi: 10.1007/s12220-017-9864-6. |
| [14] |
G. Giraud,
Sur le probléme de Dirichlet généralisé, Ann. Sci. École Norm. Sup., 46 (1929), 131-145.
|
| [15] |
Z.-C. Han and Y. Li,
The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math, J., 99 (1999), 489-542.
doi: 10.1215/S0012-7094-99-09916-7. |
| [16] |
S. Kim, M. Musso and J. Wei, Compactness of scalar-flat conformal metrics on low-dimensional manifolds manifolds with constant mean curvature on boundary, preprint, arXiv: 1906.01317. |
| [17] |
Y. Li and M. Zhu,
Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
| [18] |
F. C. Marques,
Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.
doi: 10.1512/iumj.2005.54.2590. |
| [19] |
F. C. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., 71 (2005), 315-346.
doi: 10.4310/jdg/1143651772. |
| [20] |
F. C. Marques,
Compactness and non compactness for Yamabe-type problems, Progress in Nonlinear Differential Equation and their Applications, 86 (2015), 121-131.
doi: 10.1007/978-3-319-19902-3_9. |
| [21] |
M. Mayer and C. B. Ndiaye,
Barycenter technique and the Riemann mapping problem of Cherrier-Escobar, J. Differential Geom., 107 (2017), 519-560.
doi: 10.4310/jdg/1508551224. |
| [22] |
R. Schoen and D. Zhang,
Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differ. Equ., 4 (1996), 1-25.
doi: 10.1007/BF01322307. |
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