Abstract
In this paper, we address the question of existence and uniqueness of maximizers of a class of functionals under constraints, via mass transportation theory. We also determine suitable assumptions ensuring that balls are the unique maximizers. In both cases, we show that our hypotheses are optimal.
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References
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–471 (1991)
Burton, G.R.: Vortex-rings of prescribed impulse. Math. Proc. Camb. Philos. Soc. 134(3), 515–528 (2003)
Hajaiej, H.: Balls are maximizers of the Riesz-type functionals with supermodular integrands. Accepted for publication in AMPA
Burchard, A., Hajaiej, H.: Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal. 233(2), 561–582 (2006)
Draghici, C., Hajaiej, H.: Uniqueness and characterization of maximizers of integral functionals with constraints. Adv. Nonlinear Stud. 9, 215–226 (2009)
Acknowledgements
The Author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No”RGP-VPP-124.
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Appendix: Coarea Formula
Appendix: Coarea Formula
We are changing variables through Φ:S×]0,∞[→ℝn∖{0} defined by Φ(ν,t)=νt. Let (ν,t) be fixed and let \(\{\varepsilon _{i}\}_{i=1}^{n-1}\) denote an orthonormal basis of T ν S and e a basis of T t ]0,∞[. Then {(ε i ,0),(0,e):i=1,…,n−1} is an orthonormal basis of T (ν,t)(S×]0,∞[) with
Therefore,
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Hajaiej, H. Characterization of Maximizers via Mass Transportation Techniques. J Optim Theory Appl 156, 320–329 (2013). https://doi.org/10.1007/s10957-012-0120-8
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DOI: https://doi.org/10.1007/s10957-012-0120-8