Symmetrization for Mixed Operators

[1] A. Alvino, G. Trombetti, and P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal. 13 (1989), no. 2, 185–220. AlvinoA. TrombettiG. LionsP.-L. On optimization problems with prescribed rearrangements Nonlinear Anal. 13 1989 2 185 220 Search in Google Scholar

[2] S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1611–1641. BiagiS. DipierroS. ValdinociE. VecchiE. Semilinear elliptic equations involving mixed local and nonlocal operators Proc. Roy. Soc. Edinburgh Sect. A 151 2021 5 1611 1641 Search in Google Scholar

[3] S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi, A Hong–Krahn–Szegö inequality for mixed local and nonlocal operators, Math. Eng. 5 (2022), no. 1, Paper No. 014, 25 pp. BiagiS. DipierroS. ValdinociE. VecchiE. A Hong–Krahn–Szegö inequality for mixed local and nonlocal operators Math. Eng. 5 2022 1 Paper No. 014, 25 pp Search in Google Scholar

[4] S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi, Mixed local and nonlocal elliptic operators: regularity and maximum principles, Comm. Partial Differential Equations 47 (2022), no. 3, 585–629. DOI: 10.1080/03605302.2021.1998908. BiagiS. DipierroS. ValdinociE. VecchiE. Mixed local and nonlocal elliptic operators: regularity and maximum principles Comm. Partial Differential Equations 47 2022 3 585 629 10.1080/03605302.2021.1998908 Open DOISearch in Google Scholar

[5] S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi, A Faber–Krahn inequality for mixed local and nonlocal operators, J. Anal. Math. 150 (2023), no. 2, 405–448. BiagiS. DipierroS. ValdinociE. VecchiE. A Faber–Krahn inequality for mixed local and nonlocal operators J. Anal. Math. 150 2023 2 405 448 Search in Google Scholar

[6] L. Brasco, E. Lindgren, and E. Parini, The fractional Cheeger problem, Interfaces Free Bound. 16 (2014), no. 3, 419–458. BrascoL. LindgrenE. PariniE. The fractional Cheeger problem Interfaces Free Bound. 16 2014 3 419 458 Search in Google Scholar

[7] C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal. 15 (2016), no. 2, 657–699. BucurC. Some observations on the Green function for the ball in the fractional Laplace framework Commun. Pure Appl. Anal. 15 2016 2 657 699 Search in Google Scholar

[8] D. Chen and H. Li, Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, J. Geom. Anal. 33 (2023), Paper No. 123, 20 pp. DOI: 10.1007/s12220-022-01162-0. ChenD. LiH. Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature J. Geom. Anal. 33 2023 Paper No. 123, 20 pp 10.1007/s12220-022-01162-0 Open DOISearch in Google Scholar

[9] Z.-Q. Chen, P. Kim, R. Song, and Z. Vondraček, Sharp Green function estimates for Δ+Δα2 \Delta + {\Delta^{\frac{\alpha}{2}}} in C^{1, 1} open sets and their applications, Illinois J. Math. 54 (2010), no. 3, 981–1024. ChenZ.-Q. KimP. SongR. VondračekZ. Sharp Green function estimates for Δ+Δα2 \Delta + {\Delta^{\frac{\alpha}{2}}} in C^{1, 1} open sets and their applications Illinois J. Math. 54 2010 3 981 1024

[10] Z.-Q. Chen and T. Kumagai, A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diflusions with jumps, Rev. Mat. Iberoam. 26 (2010), no. 2, 551–589. ChenZ.-Q. KumagaiT. A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diflusions with jumps Rev. Mat. Iberoam. 26 2010 2 551 589 Search in Google Scholar

[11] V. Ferone and B. Volzone, Symmetrization for fractional elliptic problems: a direct approach, Arch. Ration. Mech. Anal. 239 (2021), no. 3, 1733–1770. FeroneV. VolzoneB. Symmetrization for fractional elliptic problems: a direct approach Arch. Ration. Mech. Anal. 239 2021 3 1733 1770 Search in Google Scholar

[12] V. Ferone and B. Volzone, Symmetrization for fractional nonlinear elliptic problems, Discrete Contin. Dyn. Syst. 43 (2023), no. 3–4, 1400–1419. FeroneV. VolzoneB. Symmetrization for fractional nonlinear elliptic problems Discrete Contin. Dyn. Syst. 43 2023 3–4 1400 1419 Search in Google Scholar

[13] H. Hajaiej and K. Perera, Ground state and least positive energy solutions of elliptic problems involving mixed fractional p-Laplacians, Differential Integral Equations 35 (2022), no. 3–4, 173–190. HajaiejH. PereraK. Ground state and least positive energy solutions of elliptic problems involving mixed fractional p-Laplacians Differential Integral Equations 35 2022 3–4 173 190 Search in Google Scholar

[14] S. Kesavan, Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 453–465. KesavanS. Some remarks on a result of Talenti Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 1988 3 453 465 Search in Google Scholar

[15] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17, Acta Univ. Wratislav. No. 2029 (1997), no. 2, 339–364. KulczyckiT. Properties of Green function of symmetric stable processes Probab. Math. Statist. 17 Acta Univ. Wratislav. No. 2029 1997 2 339 364 Search in Google Scholar

[16] E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. LiebE.H. LossM. Analysis Graduate Studies in Mathematics, 14, American Mathematical Society Providence, RI 2001 Search in Google Scholar

[17] P.-L. Lions, Quelques remarques sur la symétrisation de Schwartz, in: H. Brézis, J.-L. Lions (eds.), Nonlinear Partial Diflerential Equations and Their Applications. Collège de France Seminar. Vol. I, Res. Notes in Math., 53, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981, pp. 308–319. LionsP.-L. Quelques remarques sur la symétrisation de Schwartz in: BrézisH. LionsJ.-L. (eds.), Nonlinear Partial Diflerential Equations and Their Applications. Collège de France Seminar. Vol. I Res. Notes in Math., 53, Pitman (Advanced Publishing Program) Boston, Mass.-London 1981 308 319 Search in Google Scholar

[18] P. Mironescu and W. Sickel, A Sobolev non embedding, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 3, 291–298. MironescuP. SickelW. A Sobolev non embedding Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 2015 3 291 298 Search in Google Scholar

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