Comparison results for solutions of elliptic Neumann problems with lower-order terms via Steiner symmetrization

References

[1] A. Alvino, V. Ferone and A. Mercaldo, Sharp a priori estimates for a class of nonlinear elliptic equations with lower order terms, Ann. Mat. Pura Appl. (4) 194 (2015), no. 4, 1169–1201. 10.1007/s10231-014-0416-4Search in Google Scholar

[2] A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of non-uniformly elliptic equations, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 381–391. Search in Google Scholar

[3] A. Alvino and G. Trombetti, Elliptic equations with lower-order terms and reordering (in Italian), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 66 (1979), no. 3, 194–200. Search in Google Scholar

[4] A. Alvino, G. Trombetti, J. G. Diaz and P.-I. Lions, Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math. 49 (1996), no. 3, 217–236. 10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-GSearch in Google Scholar

[5] A. Alvino, G. Trombetti and P.-L. Lions, Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 2, 37–65. 10.1016/S0294-1449(16)30303-1Search in Google Scholar

[6] A. Alvino, G. Trombetti and S. Matarasso, Elliptic boundary value problems: Comparison results via symmetrization, Ric. Mat. 51 (2002), no. 2, 341–355. Search in Google Scholar

[7] C. Bandle, Symétrisation et équations paraboliques, C.R. Acad. Sci. Paris Sér. A 280 (1975), 1113–1115. Search in Google Scholar

[8] C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math. 7, Pitman, Boston, 1980. Search in Google Scholar

[9] M. F. Betta, Neumann problem: Comparison results, Rend. Accad. Sci. Fis. Mat. Napoli (4) 57 (1990), 41–58. Search in Google Scholar

[10] M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, A comparison result related to Gauss measure, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 451–456. 10.1016/S1631-073X(02)02295-1Search in Google Scholar

[11] M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, Weighted isoperimetric inequalities on ℝn and applications to rearrangements, Math. Nachr. 281 (2008), no. 4, 466–498. 10.1002/mana.200510619Search in Google Scholar

[12] M. F. Betta and A. Mercaldo, Comparison and regularity results for a nonlinear elliptic equation, Nonlinear Anal. 20 (1993), no. 1, 63–77. 10.1016/0362-546X(93)90184-TSearch in Google Scholar

[13] L. P. Bonorino and J. B. Montenegro, Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems, Ann. Mat. Pura Appl. (4) 192 (2013), no. 6, 987–1024. 10.1007/s10231-012-0255-0Search in Google Scholar

[14] M. Bramanti, Symmetrization in parabolic Neumann problems, Appl. Anal. 40 (1991), no. 1, 21–39. 10.1080/00036819008839990Search in Google Scholar

[15] B. Brandolini, F. Chiacchio and C. Trombetti, Symmetrization for singular semilinear elliptic equations, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 389–404. 10.1007/s10231-012-0280-zSearch in Google Scholar

[16] F. Chiacchio, Comparison results for linear parabolic equations in unbounded domains via Gaussian symmetrization, Differential Integral Equations 17 (2004), no. 3–4, 241–258. 10.57262/die/1356060433Search in Google Scholar

[17] F. Chiacchio, Steiner symmetrization for an elliptic problem with lower-order terms, Ric. Mat. 53 (2004), no. 1, 87–106. Search in Google Scholar

[18] F. Chiacchio, Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization, Publ. Mat. 53 (2009), no. 1, 47–71. 10.5565/PUBLMAT_53109_02Search in Google Scholar

[19] F. Chiacchio and V. M. Monetti, Comparison results for solutions of elliptic problems via Steiner symmetrization, Differential Integral Equations 14 (2001), no. 11, 1351–1366. 10.57262/die/1356123028Search in Google Scholar

[20] A. Cianchi, Symmetrization in anisotropic elliptic problems, Comm. Partial Differential Equations 32 (2007), no. 4–6, 693–717. 10.1080/03605300600634973Search in Google Scholar

[21] A. Cianchi and V. Maz’ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 3, 571–595. 10.4171/JEMS/440Search in Google Scholar

[22] E. De Giorgi Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura. Appl. 36 (1954), 191–213. 10.1007/BF02412838Search in Google Scholar

[23] J. I. Díaz and D. Gómez-Castro, Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem, Discrete Contin. Dyn. Syst. 2015 (2015), 379–386. 10.3934/proc.2015.0379Search in Google Scholar

[24] G. Di Blasio and B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations 253 (2012), no. 9, 2593–2615. 10.1016/j.jde.2012.07.004Search in Google Scholar

[25] J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Funct. Anal. 257 (2009), no. 3, 807–831. 10.1016/j.jfa.2009.03.002Search in Google Scholar

[26] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458–520. 10.2307/1970227Search in Google Scholar

[27] A. Ferone, Symmetrization for degenerate Neumann problems, Rend. Accad. Sci. Fis. Mat. Napoli (4) 60 (1993), 27–46. Search in Google Scholar

[28] V. Ferone, Symmetrization in a Neumann problem, Matematiche (Catania) 41 (1986), no. 1–2, 67–78. Search in Google Scholar

[29] V. Ferone, Isoperimetric inequalities and applications (in Italian), Boll. Unione Mat. Ital. (9) 1 (2008), no. 3, 539–557. Search in Google Scholar

[30] V. Ferone and A. Mercaldo, Neumann problems and Steiner symmetrization, Comm. Partial Differential Equations 30 (2005), no. 10–12, 1537–1553. 10.1080/03605300500299596Search in Google Scholar

[31] V. Ferone and B. Messano, Comparison results for nonlinear elliptic equations with lower-order terms, Math. Nachr. 252 (2003), 43–50. 10.1002/mana.200310036Search in Google Scholar

[32] V. Ferone and B. Messano, A symmetrization result for nonlinear elliptic equations, Rev. Mat. Complut. 17 (2004), no. 2, 261–276. 10.5209/rev_REMA.2004.v17.n2.16718Search in Google Scholar

[33] V. Ferone and B. Messano, Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient, Adv. Nonlinear Stud. 7 (2007), no. 1, 31–46. 10.1515/ans-2007-0102Search in Google Scholar

[34] V. Ferone and F. Murat, Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces, J. Differential Equations 256 (2014), no. 2, 577–608. 10.1016/j.jde.2013.09.013Search in Google Scholar

[35] G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. Search in Google Scholar

[36] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer, Berlin, 1985. 10.1007/BFb0075060Search in Google Scholar

[37] S. Kesavan, Symmetrization and Applications, Ser. Anal. 3, World Scientific, Hackensack, 2006. 10.1142/6071Search in Google Scholar

[38] M. A. Krasnosel’skiĭ, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964. Search in Google Scholar

[39] J. J. Langford, Neumann comparison theorems in elliptic PDEs, Potential Anal. 43 (2015), no. 3, 415–459. 10.1007/s11118-015-9479-0Search in Google Scholar

[40] W. B. Li and F. Q. Li, Comparison results for solutions of sublinear elliptic equations via Steiner symmetrization, Complex Var. Elliptic Equ. 61 (2016), no. 2, 213–224. 10.1080/17476933.2015.1073272Search in Google Scholar

[41] C. Maderna and S. Salsa, Symmetrization in Neumann problems, Appl. Anal. 9 (1979), no. 4, 247–256. 10.1080/00036817908839273Search in Google Scholar

[42] V. G. Maz’ja, Weak solutions of the Dirichlet and Neumann problems (in Russian), Tr. Mosk. Mat. Obs. 20 (1969), 137–172. Search in Google Scholar

[43] B. Messano, Symmetrization results for classes of nonlinear elliptic equations with q-growth in the gradient, Nonlinear Anal. 64 (2006), no. 12, 2688–2703. 10.1016/j.na.2005.07.042Search in Google Scholar

[44] J. Mossino and J.-M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 1, 51–73. Search in Google Scholar

[45] J.-M. Rakotoson, Réarrangement Relatif. Un instrument d’estimations dans les problèmes aux limites, Math. Appl. (Berlin) 64, Springer, Berlin, 2008. Search in Google Scholar

[46] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189–258. 10.5802/aif.204Search in Google Scholar

[47] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 697–718. Search in Google Scholar

[48] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 160–184. 10.1007/BF02411942Search in Google Scholar

[49] Y. J. Tian and F. Q. Li, Comparison results for nonlinear degenerate Dirichlet and Neumann problems with general growth in the gradient, J. Math. Anal. Appl. 378 (2011), no. 2, 749–763. 10.1016/j.jmaa.2011.01.072Search in Google Scholar

[50] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Univ. Ser. Math., Plenum Press, New York, 1987. 10.1007/978-1-4899-3614-1Search in Google Scholar

[51] G. Trombetti, Symmetrization methods for partial differential equations (in Italian), Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), no. 3, 601–634. Search in Google Scholar

[52] J. L. Vázquez, Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations, Adv. Nonlinear Stud. 5 (2005), no. 1, 87–131. 10.1515/ans-2005-0107Search in Google Scholar

[53] 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), no. 5, 553–582. 10.1016/j.matpur.2013.07.001Search in Google Scholar