Abstract
The talk given on the occasion of the ISIAM 2016 conference was mainly about rigidity results for nonnegative solutions of semilinear elliptic equation on infinite cylinder-like domains or in the Euclidean espace and as a consequence, about optimal symmetry properties for the optimizers of the Caffarelli–Kohn–Nirenberg inequalities. This text contains the main results presented in that conference. All the results will be stated in the simple case of spherical cylinders, but similar, even if less precise, results can also be stated and proved for general cylinders generated by any compact smooth Riemannian manifold without a boundary. Other consequences from the results below are optimal estimates for the principal eigenvalue of Schrödinger operators on infinite cylinders. The text below is an extended abstract of that talk.
Work done in collaboration with J. Dolbeault and M. Loss.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53(3), 259–275 (1984)
Catrina, F., Wang, Z.-Q.: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 54(2), 229–258 (2001)
Dolbeault, J., Esteban, M.J., Loss, M.: Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. Invent. Math. 206(2), 397–440 (2016)
Felli, V., Schneider, M.: Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type. J. Differ. Equs. 191(1), 121–142 (2003)
Vázquez, J.L.: Asymptotic behaviour for the porous medium equation posed in the whole space. Nonlinear Evolution Equations and Related Topics, pp. 67–118. Springer, Berlin (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Esteban, M.J. (2017). Nonlinear Flows and Optimality for Functional Inequalities: An Extended Abstract. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Industrial Mathematics and Complex Systems. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3758-0_2
Download citation
DOI: https://doi.org/10.1007/978-981-10-3758-0_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3757-3
Online ISBN: 978-981-10-3758-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)