Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

Marco G. Ghimenti; Anna Maria Micheletti; Marco G. Ghimenti; Anna Maria Micheletti

doi:10.3934/era.2022064

Electronic Research Archive

2022, Volume 30, Issue 4: 1209-1235. doi: 10.3934/era.2022064

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Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

Marco G. Ghimenti ,
Anna Maria Micheletti ^,

Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56126 Pisa, Italy

Received: 07 July 2021 Revised: 30 December 2021 Accepted: 30 December 2021 Published: 14 March 2022

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.
- non umbilic boundary,
- Yamabe problem,
- compactness,
- blow up analysis
Citation: Marco G. Ghimenti, Anna Maria Micheletti. Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary[J]. Electronic Research Archive, 2022, 30(4): 1209-1235. doi: 10.3934/era.2022064

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Abstract

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.

References

[1]	O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 23 (2004), 1143–1191. https://doi.org/10.1155/S1073792804133278 doi: 10.1155/S1073792804133278
[2]	J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. Math., 136 (1992), 1–50. https://doi.org/10.2307/2946545 doi: 10.2307/2946545
[3]	F. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599–1620. https://doi.org/10.1512/iumj.2005.54.2590 doi: 10.1512/iumj.2005.54.2590
[4]	S. Almaraz, An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1–22. https://doi.org/10.2140/pjm.2010.248.1 doi: 10.2140/pjm.2010.248.1
[5]	S. Brendle, S. S. Chen, An existence theorem for the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc., 16 (2014), 991–1016. https://doi.org/10.4171/JEMS/453 doi: 10.4171/JEMS/453
[6]	M. Mayer, C. B. Ndiaye, Barycenter technique and the Riemann mapping problem of Cherrier-Escobar, J. Differ. Geom., 107 (2017), 519–560. https://doi.org/10.4310/jdg/1508551224 doi: 10.4310/jdg/1508551224
[7]	M. Khuri, F. Marques, R. Schoen, A compactness theorem for the Yamabe problem, J. Differ. Geom., 81 (2009), 143–196. https://doi.org/10.4310/jdg/1228400630 doi: 10.4310/jdg/1228400630
[8]	V. Felli, M. Ould Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175–210. https://doi.org/10.1007/s00209-002-0486-7 doi: 10.1007/s00209-002-0486-7
[9]	S. Almaraz, A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var., 41 (2011), 341–386. https://doi.org/10.1007/s00526-010-0365-8 doi: 10.1007/s00526-010-0365-8
[10]	M. G. Ghimenti, A. M. Micheletti, Compactness for conformal scalar-flat metrics on umbilic boundary manifolds, Nonlinear Anal., 200 (2020). https://doi.org/10.1016/j.na.2020.111992
[11]	M. Ghimenti, A. M. Micheletti, A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary, in press on Calc Var PDE.
[12]	S. Almaraz, Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differ. Equ., 251 (2011), 1813–1840. https://doi.org/10.1016/j.jde.2011.04.013 doi: 10.1016/j.jde.2011.04.013
[13]	O. Druet, E. Hebey, Sharp asymptotics and compactness for local low energy solutions of critical elliptic systems in potential form, Calc. Var. Partial Differ. Equ., 31 (2008), 205–230. https://doi.org/10.1007/s00526-007-0111-z doi: 10.1007/s00526-007-0111-z
[14]	O. Druet, E. Hebey, F. Robert, A C0-theory for the blow-up of second order elliptic equations of critical Sobolev growth, Elect. Res. Ann. A.M.S, 9 (2003), 19–25. https://doi.org/10.1090/S1079-6762-03-00108-2 doi: 10.1090/S1079-6762-03-00108-2
[15]	M. G. Ghimenti, A. M. Micheletti, Compactness results for linearly perturbed Yamabe problem on manifolds with boundary, Discrete Contin Dyn Syst, series S, 14 (2021), 1757–1778. https://doi.org/10.3934/dcdss.2020453
[16]	M. G. Ghimenti, A. M. Micheletti, A. Pistoia, Linear Perturbation of the Yamabe Problem on Manifolds with Boundary, J. Geom. Anal., 28 (2018), 1315–1340 https://doi.org/10.1007/s12220-017-9864-6 doi: 10.1007/s12220-017-9864-6
[17]	P. Esposito, A. Pistoia, J. Vetois, The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511–560. https://doi.org/10.1007/s00208-013-0971-9 doi: 10.1007/s00208-013-0971-9
[18]	A. M. Micheletti, A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds, Calc. Var. Partial Differ. Equ., 34 (2009), 233–265. https://doi.org/10.1007/s00526-008-0183-4 doi: 10.1007/s00526-008-0183-4
[19]	Z. C. Han, Y. Li, The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math, J., 99 (1999), 489–542. https://doi.org/10.1215/S0012-7094-99-09916-7 doi: 10.1215/S0012-7094-99-09916-7
[20]	F. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., 71 (2005), 315–346. https://doi.org/10.4310/jdg/1143651772 doi: 10.4310/jdg/1143651772
[21]	Y. Li, M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1–50. https://doi.org/10.1142/S021919979900002X doi: 10.1142/S021919979900002X
[22]	R. Schoen, D. Zhang, Prescribed scalar curvature on the n-sphere, Calc. Var. Partial Differ. Equ., 4 (1996), 1–25. https://doi.org/10.1007/BF01322307
[23]	T. Aubin, Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin, 1998. https://doi.org/10.1007/978-3-662-13006-3
[24]	G. Giraud, Sur la problème de Dirichlet généralisé, Ann. Sci.École Norm. Sup., 46 (1929), 131–145. https://doi.org/10.24033/asens.793 doi: 10.24033/asens.793
[25]	W. M. Ni, I. Takagi, On the Shape of Least-Energy Solutions to a Semilinear Neumann Problem, Comm. Pure Appl. Math., 44 (1991), 819–851. https://doi.org/10.1002/cpa.3160440705 doi: 10.1002/cpa.3160440705

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