Optimization problems in rearrangement classes for fractional $ p $-Laplacian equations

Antonio Iannizzotto; Giovanni Porru; Antonio Iannizzotto; Giovanni Porru

doi:10.3934/mine.2025002

Mathematics in Engineering

2025, Volume 7, Issue 1: 13-34. doi: 10.3934/mine.2025002

Previous Article Next Article

Research article

Optimization problems in rearrangement classes for fractional $ p $-Laplacian equations

Antonio Iannizzotto ^,,
Giovanni Porru

Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy

Received: 18 November 2024 Revised: 25 January 2025 Accepted: 05 February 2025 Published: 13 February 2025

We discuss two optimization problems related to the fractional $ p $-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $ p $-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the minimization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution.
- rearrangement class,
- fractional $ p $-Laplacian,
- eigenvalues
Citation: Antonio Iannizzotto, Giovanni Porru. Optimization problems in rearrangement classes for fractional $ p $-Laplacian equations[J]. Mathematics in Engineering, 2025, 7(1): 13-34. doi: 10.3934/mine.2025002

Related Papers:

Abstract

We discuss two optimization problems related to the fractional $ p $-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $ p $-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the minimization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution.

References

[1]	C. Anedda, F. Cuccu, S. Frassu, Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight, Can. J. Math., 73 (2021), 970–992. https://doi.org/10.4153/S0008414X20000267 doi: 10.4153/S0008414X20000267
[2]	C. Bjorland, L. Caffarelli, A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337–380. https://doi.org/10.1002/cpa.21379 doi: 10.1002/cpa.21379
[3]	G. R. Burton, Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., 276 (1987), 225–253. https://doi.org/10.1007/BF01450739 doi: 10.1007/BF01450739
[4]	G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincaré, 6 (1989), 295–319. https://doi.org/10.1016/S0294-1449(16)30320-1 doi: 10.1016/S0294-1449(16)30320-1
[5]	G. R. Burton, J. B. McLeod, Maximisation and minimisation on classes of rearrangements, Proc. R. Soc. Edinburgh: Sec. A Math., 119 (1991), 287–300. https://doi.org/10.1017/S0308210500014840 doi: 10.1017/S0308210500014840
[6]	L. Caffarelli, Non-local diffusions, drifts and games, In: H. Holden, K. Karlsen, Nonlinear partial differential equations, Abel Symposia, Springer, Berlin, 7 (2012), 37–52. https://doi.org/10.1007/978-3-642-25361-4_3
[7]	F. Cuccu, B. Emamizadeh, G. Porru, Optimization of the first eigenvalue in problems involving the $p$-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 1677–1687.
[8]	F. Cuccu, G. Porru, S. Sakaguchi, Optimization problems on general classes of rearrangements, Nonlinear Anal., 74 (2011), 5554–5565. https://doi.org/10.1016/j.na.2011.05.039 doi: 10.1016/j.na.2011.05.039
[9]	L. M. Del Pezzo, A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian, J. Differ. Equ., 263 (2017), 765–778. https://doi.org/10.1016/j.jde.2017.02.051 doi: 10.1016/j.jde.2017.02.051
[10]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[11]	G. Franzina, G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373–386.
[12]	A. Iannizzotto, Monotonicity of eigenvalues of the fractional $p$-Laplacian with singular weights, Topol. Methods Nonlinear Anal., 61 (2023), 423–443. https://doi.org/10.12775/TMNA.2022.024 doi: 10.12775/TMNA.2022.024
[13]	A. Iannizzotto, A survey on boundary regularity for the fractional $p$-Laplacian and its applications, Bruno Pini Math. Anal. Seminar, 15 (2024), 164–186.
[14]	A. Iannizzotto, S. Liu, K. Perera, M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101–125. https://doi.org/10.1515/acv-2014-0024 doi: 10.1515/acv-2014-0024
[15]	A. Iannizzotto, S. Mosconi, Fine boundary regularity for the singular fractional $p$-Laplacian, J. Differ. Equations, 412 (2024), 322–379. https://doi.org/10.1016/j.jde.2024.08.026 doi: 10.1016/j.jde.2024.08.026
[16]	A. Iannizzotto, S. Mosconi, On a doubly sublinear fractional $p$-Laplacian equation, arXiv, 2024. https://doi.org/10.48550/arXiv.2409.03616
[17]	A. Iannizzotto, S. Mosconi, N. S. Papageorgiou, On the logistic equation for the fractional $p$-Laplacian, Math. Nachr., 296 (2023), 1451–1468. https://doi.org/10.1002/mana.202100025 doi: 10.1002/mana.202100025
[18]	A. Iannizzotto, D. Mugnai, Optimal solvability for the fractional $p$-Laplacian with Dirichlet conditions, Fract. Calc. Appl. Anal., 27 (2024), 3291–3317. https://doi.org/10.1007/s13540-024-00341-w doi: 10.1007/s13540-024-00341-w
[19]	H. Ishii, G. Nakamura, A class of integral equations and approximation of $p$-Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485–522. https://doi.org/10.1007/s00526-009-0274-x doi: 10.1007/s00526-009-0274-x
[20]	B. Kawohl, M. Lucia, S. Prashanth, Simplicity of the first eigenvalue for indefinite quasilinear problems, Adv. Differ. Equations, 12 (2007), 407–434. https://doi.org/10.57262/ade/1355867457 doi: 10.57262/ade/1355867457
[21]	G. Leoni, A first course in fractional Sobolev spaces, Vol. 229, American Mathematical Society, 2023.
[22]	E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826. https://doi.org/10.1007/s00526-013-0600-1 doi: 10.1007/s00526-013-0600-1
[23]	G. Molica Bisci, V. D. Rădulescu, R. Servadei, Variational methods for nonlocal fractional problems, Cambridge: Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316282397
[24]	G. Palatucci, The Dirichlet problem for the $p$-fractional Laplace equation, Nonlinear Anal., 177 (2018), 699–732. https://doi.org/10.1016/j.na.2018.05.004 doi: 10.1016/j.na.2018.05.004
[25]	B. Pellacci, G. Verzini, Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems, J. Math. Biol., 76 (2018), 1357–1386. https://doi.org/10.1007/s00285-017-1180-z doi: 10.1007/s00285-017-1180-z
[26]	C. Qiu, Y. Huang, Y. Zhou, Optimization problems involving the fractional Laplacian, Electron. J. Differ. Eq., 2016 (2016), 1–15.
[27]	T. R. Rockafellar, Convex analysis, Princeton University Press, 1970.
[28]	X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3–26.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)