We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal.239 (2021) 1733–1770, Theorem 31). Some interesting consequences are regularity results and nonlocal energy estimates for solutions.
F.J.Almgren and E.H.Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc.2 (1989), 683–773. doi:10.1090/S0894-0347-1989-1002633-4.
2.
A.Alvino, P.-L.Lions and G.Trombetti, On optimization problems with prescribed rearrangements, Nonlinear Anal.13 (1989), 185–220. doi:10.1016/0362-546X(89)90043-6.
3.
A.Alvino, R.Volpicelli and B.Volzone, Sharp estimates for solutions of parabolic equations with a lower order term, J. Appl. Funct. Anal.3 (2008), 61–88.
4.
C.Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, Vol. 7, Pitman (Advanced Publishing Program), Boston, MA, 1980.
5.
B.Barrios, I.De Bonis, M.Medina and I.Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math.13 (2015), 390–407.
6.
C.Bennett and R.Sharpley, Interpolation of Operators, Pure and App. Math., Vol. 129, Academic Press Inc., Boston, MA, 1988.
7.
L.Boccardo and L.Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations37 (2010), 363–380. doi:10.1007/s00526-009-0266-x.
8.
M.Bonforte, Y.Sire and J.L.Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst.35 (2015), 5725–5767. doi:10.3934/dcds.2015.35.5725.
9.
B.Brandolini, F.Chiacchio and C.Trombetti, Symmetrization for singular semilinear elliptic equations, Ann. Mat. Pura Appl. (4)193 (2014), 389–404. doi:10.1007/s10231-012-0280-z.
10.
L.Brasco, E.Lindgren and E.Parini, The fractional Cheeger problem, Interfaces Free Bound.16 (2014), 419–458. doi:10.4171/IFB/325.
11.
L.Brasco and E.Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var.9 (2016), 323–355. doi:10.1515/acv-2015-0007.
12.
L.Brasco, E.Parini and M.Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst.36 (2016), 1813–1845. doi:10.3934/dcds.2016.36.1813.
13.
A.Canino, L.Montoro, B.Sciunzi and M.Squassina, Nonlocal problems with singular nonlinearity, Bull. Sci. Math.141 (2017), 223–250. doi:10.1016/j.bulsci.2017.01.002.
14.
G.di Blasio and B.Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations253 (2012), 2593–2615. doi:10.1016/j.jde.2012.07.004.
15.
E.Di Nezza, G.Palatucci and E.Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.136 (2012), 521–573. doi:10.1016/j.bulsci.2011.12.004.
16.
P.Felmer and Y.Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math.16 (2014), 1350023.
17.
F.Feo, P.R.Stinga and B.Volzone, The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity, Discrete Contin. Dyn. Syst.38 (2018), 3269–3298. doi:10.3934/dcds.2018142.
18.
V.Ferone and B.Volzone, Symmetrization for fractional elliptic problems: A direct approach, Arch. Ration. Mech. Anal.239 (2021), 1733–1770. doi:10.1007/s00205-020-01601-8.
19.
V.Ferone and B.Volzone, Symmetrization for fractional nonlinear elliptic problems, Discrete Contin. Dyn. Syst. Ser. A43 (2023), 1400–1419. doi:10.3934/dcds.2022076.
20.
G.Galiano, Symmetrization in nonlocal diffusion problems, 2022, arXiv:2208.14735.
21.
S.Kesavan, Symmetrization & Applications, Series in Analysis, Vol. 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
22.
N.N.Lebedev, Special Functions and Their Applications, Revised edn, Dover Publications, Inc., New York, 1972, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication.
23.
T.Leonori, I.A.Peral, A.Primo and F.Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst.35 (2015), 6031–6068. doi:10.3934/dcds.2015.35.6031.
24.
X.Ros-Oton and J.Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl.101 (2014), 275–302. doi:10.1016/j.matpur.2013.06.003.
25.
X.Ros-Oton and J.Serra, Regularity theory for general stable operators, J. Differential equations260 (2016), 8675–8715. doi:10.1016/j.jde.2016.02.033.
26.
Y.Sire, J.L.Vázquez and B.Volzone, Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application, Chinese Ann. Math. Ser. B38 (2017), 661–686. doi:10.1007/s11401-017-1089-2.
27.
E.M.Stein, Singular Integrals and Differential Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N. J., 1970.
28.
G.Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)3 (1976), 697–718.
29.
G.Talenti, Inequalities in rearrangement invariant function spaces, in: Nonlinear Analysis, Function Spaces and Applications, Prague, 1994, Vol. 5, Prometheus, Prague, 1994, pp. 177–230.
30.
G.Talenti, The art of rearranging, Milan J. Math.84 (2016), 105–157. doi:10.1007/s00032-016-0253-6.
31.
J.L.Vázquez and B.Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl. (9)101 (2014), 553–582. doi:10.1016/j.matpur.2013.07.001.
32.
J.L.Vázquez and B.Volzone, Optimal estimates for fractional fast diffusion equations, J. Math. Pures Appl. (9)103 (2015), 535–556. doi:10.1016/j.matpur.2014.07.002.
