Open Access
Volume 29, 2023
Article Number 28
Number of page(s) 28
Published online 26 April 2023
  1. E. Abreu and L.G. Fernandes, Jr., On the existence and nonexistence of isoperimetric inequalities with different monomial weights. J. Fourier Anal. Appl. 28 (2022) 33. [Google Scholar]
  2. W.K. Allard, A regularity theorem for the first variation of the area integrand. Bull. Am. Math. Soc. 77 (1971) 772–776. [Google Scholar]
  3. W.K. Allard, On the first variation of a varifold. Ann. Math. 95 (1972) 417–491. [Google Scholar]
  4. W.K. Allard, On the first variation of a varifold: boundary behavior. Ann. Math. 101 (1975) 418–446. [Google Scholar]
  5. F.J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4 (1976) viii+199. [Google Scholar]
  6. F.J. Almgren, Jr., Almgren’s big regularity paper, Vol. 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ (2000). [Google Scholar]
  7. A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo and M.R. Posteraro, Some isoperimetric inequalities on ℝN with respect to weights |x|α. J. Math. Anal. Appl. 451 (2017) 280–318. [Google Scholar]
  8. A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo and M.R. Posteraro, On weighted isoperimetric inequalities with non-radial densities. Appl. Anal. 98 (2019) 1935–1945. [Google Scholar]
  9. A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo and M.R. Posteraro, Some isoperimetric inequalities with respect to monomial weights. ESAIM Control Optim. Calc. Var. 27 (2021) 29. [Google Scholar]
  10. A. Alvino, V. Ferone and C. Nitsch. A sharp isoperimetric inequality in the plane. J. Eur. Math. Soc. 13 (2011) 185–206. [Google Scholar]
  11. Z.M. Balogh, C.E. Gutiérrez and A. Kristály, Sobolev inequalities with jointly concave weights on convex cones. Proc. Lond. Math. Soc. 122 (2021) 537–568. [Google Scholar]
  12. M. Barchiesi, F. Cagnetti and N. Fusco, Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013) 1245–1278. [Google Scholar]
  13. M.F. Betta, F. Brock, A. Mercaldo and M.R. Posteraro, A weighted isoperimetric inequality and applications to symmetrization. J. Inequal. Appl. 4 (1999) 215–240. [Google Scholar]
  14. M.F. Betta, F. Brock, A. Mercaldo and M.R. Posteraro, Weighted isoperimetric inequalities on ℝn and applications to rearrangements. Math. Nachr. 281 (2008) 466–498. [Google Scholar]
  15. G. Bianchi, R.J. Gardner and P. Gronchi, Symmetrization in geometry. Adv. Math. 306 (2017) 51–88. [Google Scholar]
  16. E. Bombieri, Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99–130. [Google Scholar]
  17. W. Boyer, B. Brown, G.R. Chambers, A. Loving and S. Tammen, Isoperimetric regions in ℝn with density rp. Anal. Geom. Metr. Spaces 4 (2016) 236–265. [MathSciNet] [Google Scholar]
  18. F. Brock, A. Mercaldo and M.R. Posteraro, On isoperimetric inequalities with respect to infinite measures. Rev. Mat. Iberoam. 29 (2013) 665–690. [CrossRef] [MathSciNet] [Google Scholar]
  19. F. Brock and A.Y. Solynin, An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352 (2000) 1759–1796. [Google Scholar]
  20. X. Cabré, Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: a survey. Chin. Ann. Math. Ser. B 38 (2017) 201–214. [CrossRef] [Google Scholar]
  21. X. Cabré and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights. J. Diff. Equ. 255 (2013) 4312–4336. [Google Scholar]
  22. X. Cabré, X. Ros-Oton, and J. Serra. Sharp isoperimetric inequalities via the ABP method. J. Eur. Math. Soc. (JEMS) 18 (2016) 2971–2998. [CrossRef] [MathSciNet] [Google Scholar]
  23. F. Cagnetti, M. Perugini and D. Stöger, Rigidity for perimeter inequality under spherical symmetrisation. Calc. Var. Part. Diff. Equ. 59 (2020) 53. [Google Scholar]
  24. C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density. Bull. Aust. Math. Soc. 78 (2008) 177–197. [Google Scholar]
  25. M. Chlebík, A. Cianchi and N. Fusco, The perimeter inequality under Steiner symmetrization: cases of equality. Ann. Math. 162 (2005) 525–555. [CrossRef] [MathSciNet] [Google Scholar]
  26. E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton and J. Serra, Sharp quantitative stability for isoperimetric inequalities with homogeneous weights. Trans. Am. Math. Soc. 375 (2022) 1509–1550. [CrossRef] [Google Scholar]
  27. E. Cinti and A. Pratelli, Regularity of isoperimetric sets in ℝ2 with density. Math. Ann. 368 (2017) 419–432. [CrossRef] [MathSciNet] [Google Scholar]
  28. E. Cinti and A. Pratelli, The ɛ — ɛβ property, the boundedness of isoperimetric sets in ℝN with density, and some applications. J. Reine Angew. Math. 728 (2017) 65–103. [Google Scholar]
  29. G. Csató, An isoperimetric problem with density and the Hardy Sobolev inequality in ℝ2. Diff. Integral Equ. 28 (2015) 971–988. [Google Scholar]
  30. G. Csató, On the isoperimetric problem with perimeter density rp. Commun. Pure Appl. Anal. 17 (2018) 2729–2749. [Google Scholar]
  31. J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density rp. New York J. Math. 16 (2010) 31–51. [MathSciNet] [Google Scholar]
  32. L. Di Giosia, J. Habib, L. Kenigsberg, D. Pittman and W. Zhu. Balls Isoperimetric in ℝn with Volume and Perimeter Densities rm and rk, October 2016. [Google Scholar]
  33. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Textbooks in Mathematics, revised edn. CRC Press, Boca Raton, FL (2015). [Google Scholar]
  34. H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). [Google Scholar]
  35. N. Fusco, F. Maggi and A. Pratelli, On the isoperimetric problem with respect to a mixed Euclidean—Gaussian density. J. Funct. Anal. 260 (2011) 3678–3717. [Google Scholar]
  36. B. Kawohl, Rearrangements and Convexity of Level Sets in PDE. Vol. 1150 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1985). [Google Scholar]
  37. S. Kesavan, Symmetrization & Applications. Vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). [CrossRef] [Google Scholar]
  38. F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. Vol. 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012). [Google Scholar]
  39. F. Morgan, Geometric Measure Theory, 5th edn. Elsevier/Academic Press, Amsterdam (2016). A beginner’s guide, Illustrated by James F. Bredt. [Google Scholar]
  40. F. Morgan and A. Pratelli, Existence of isoperimetric regions in ℝn with density. Ann. Global Anal. Geom. 43 (2013) 331–365. [CrossRef] [MathSciNet] [Google Scholar]
  41. A. Pratelli and G. Saracco, The ɛ — ɛβ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets. Adv. Nonlinear Stud. 20 (2020) 539–555. [CrossRef] [MathSciNet] [Google Scholar]
  42. C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density. Calc. Var. Part. Diff. Equ. 31 (2008) 27–46. [Google Scholar]
  43. A. I. Vol’pert, Spaces BV and quasilinear equations. Mat. Sb. (N.S.) 73 (1967) 255–302. [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.