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November  2018, 17(6): 2729-2749. doi: 10.3934/cpaa.2018129

On the isoperimetric problem with perimeter density $r^p$

Gyula Csató ,

Universitat Politècnica de Catalunya, member of BGSMath, Barcelona, Spain

* Corresponding author

Received  December 2016 Revised  July 2017 Published  June 2018

Fund Project: The author is supported by FONDECYT grant 11150017.

In this paper the author studies the isoperimetric problem in ${\mathbb{R}}^n$ with perimeter density $|x|^p$ and volume density 1. We settle completely the case $n = 2$, completing a previous work by the author: we characterize the case of equality if $0≤p≤1$ and deal with the case $-∞<p<-1$ (with the additional assumption $0∈Ω$). In the case $n≥3$ we deal mainly with the case $-∞<p<0$, showing among others that the results in 2 dimensions do not generalize for the range $-n+1<p<0.$

Keywords: Isoperimetric problem with density, radial weight, case of equality, first variation, starshaped domains.
Mathematics Subject Classification: Primary: 49Q10, 49Q20; Secondary: 26D10.
Citation: Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
References:
[1]

A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo and M. R. Posteraro, Some isoperimetric inequalities on ${\mathbb{R}}^n$ with respect to weights $|x|^{α}$, J. Math. Anal. Appl., 1 (2017), 280-318.  doi: 10.1016/j.jmaa.2017.01.085.  Google Scholar

[2]

A. Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[3]

M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, A weighted isoperimetric inequality and applications to symmetrization, J. of Inequal. and Appl., 4 (1999), 215-240.  doi: 10.1155/S1025583499000375.  Google Scholar

[4]

W. Boyer, B. Brown, G. R. Chambers, A. Loving and S. Tammen, Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$, Anal. Geom. Metr. Spaces, 4 (2016), 236-265.  doi: 10.1515/agms-2016-0009.  Google Scholar

[5]

V. Bayle, A. Cañete, F. Morgan and C. Rosales, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differential Equations, 31 (2008), 27-46.  doi: 10.1007/s00526-007-0104-y.  Google Scholar

[6]

X. Cabré and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations, 255 (2013), 4312-4336.  doi: 10.1016/j.jde.2013.08.010.  Google Scholar

[7]

X. Cabré, X. Ros-Oton and J. Serra, Euclidean balls solve some isoperimetric problems with nonradial weights, C. R. Math. Acad. Sci. Paris, 350 (2012), 945-947.  doi: 10.1016/j.crma.2012.10.031.  Google Scholar

[8]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geom. Anal., 20 (2010), 243-290.  doi: 10.1007/s12220-009-9109-4.  Google Scholar

[9]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197.  doi: 10.1017/S000497270800052X.  Google Scholar

[10]

G. R. Chambers, Proof of the log-convex density conjecture, J. Eur. Math. Soc., to appear. Google Scholar

[11]

G. Csató, An isoperimetric problem with density and the Hardy-Sobolev inequality in ${\mathbb{R}}^2$, Differential Integral Equations, 28 (2015), 971-988.   Google Scholar

[12]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54 (2015), 2341-2366.  doi: 10.1007/s00526-015-0867-5.  Google Scholar

[13]

G. Csató and P. Roy, The singular Moser-Trudinger inequality on simply connected domains, Communications in Partial Differential Equations, 41 (2016), 838-847.  doi: 10.1080/03605302.2015.1123276.  Google Scholar

[14]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010), 31-51.   Google Scholar

[15]

A. Díaz, N. Harman, S. Howe and D. Thompson, Isoperimetric problems in sectors with density, Adv. Geom., 12 (2012), 589-619.   Google Scholar

[16]

J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z., 185 (1984), 339-353.  doi: 10.1007/BF01215045.  Google Scholar

[17]

A. Figalli and F. Maggi, On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations, 48 (2013), 447-489.  doi: 10.1007/s00526-012-0557-5.  Google Scholar

[18]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497.  doi: 10.1007/BF02566514.  Google Scholar

[19]

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.  doi: 10.1016/j.jfa.2011.01.007.  Google Scholar

[20]

L. Di Giosia, J. Habib, L. Kenigsberg, D. Pittman and W. Zhu, Balls Isoperimetric in ${\mathbb{R}}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$, preprint, arXiv: 1610.05830v1. Google Scholar

[21]

F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc., 355 (2003), 5041-5052.  doi: 10.1090/S0002-9947-03-03061-7.  Google Scholar

[22]

F. Morgan, Available from: http://sites.williams.edu/Morgan/2010/06/22/variation-formulae-for-perimeter-and-volume-densities/. Google Scholar

[23]

F. Morgan and A. Pratelli, Existence of isoperimetric regions in $\mathbb{R}^n$ with density, Ann. Global Anal. Geom., 43 (2013), 331-365.  doi: 10.1007/s10455-012-9348-7.  Google Scholar

[24]

W. Walter, Ordinary Differential Equations, English translation, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.  Google Scholar

show all references

References:
[1]

A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo and M. R. Posteraro, Some isoperimetric inequalities on ${\mathbb{R}}^n$ with respect to weights $|x|^{α}$, J. Math. Anal. Appl., 1 (2017), 280-318.  doi: 10.1016/j.jmaa.2017.01.085.  Google Scholar

[2]

A. Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar

[3]

M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, A weighted isoperimetric inequality and applications to symmetrization, J. of Inequal. and Appl., 4 (1999), 215-240.  doi: 10.1155/S1025583499000375.  Google Scholar

[4]

W. Boyer, B. Brown, G. R. Chambers, A. Loving and S. Tammen, Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$, Anal. Geom. Metr. Spaces, 4 (2016), 236-265.  doi: 10.1515/agms-2016-0009.  Google Scholar

[5]

V. Bayle, A. Cañete, F. Morgan and C. Rosales, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differential Equations, 31 (2008), 27-46.  doi: 10.1007/s00526-007-0104-y.  Google Scholar

[6]

X. Cabré and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations, 255 (2013), 4312-4336.  doi: 10.1016/j.jde.2013.08.010.  Google Scholar

[7]

X. Cabré, X. Ros-Oton and J. Serra, Euclidean balls solve some isoperimetric problems with nonradial weights, C. R. Math. Acad. Sci. Paris, 350 (2012), 945-947.  doi: 10.1016/j.crma.2012.10.031.  Google Scholar

[8]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geom. Anal., 20 (2010), 243-290.  doi: 10.1007/s12220-009-9109-4.  Google Scholar

[9]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197.  doi: 10.1017/S000497270800052X.  Google Scholar

[10]

G. R. Chambers, Proof of the log-convex density conjecture, J. Eur. Math. Soc., to appear. Google Scholar

[11]

G. Csató, An isoperimetric problem with density and the Hardy-Sobolev inequality in ${\mathbb{R}}^2$, Differential Integral Equations, 28 (2015), 971-988.   Google Scholar

[12]

G. Csató and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54 (2015), 2341-2366.  doi: 10.1007/s00526-015-0867-5.  Google Scholar

[13]

G. Csató and P. Roy, The singular Moser-Trudinger inequality on simply connected domains, Communications in Partial Differential Equations, 41 (2016), 838-847.  doi: 10.1080/03605302.2015.1123276.  Google Scholar

[14]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010), 31-51.   Google Scholar

[15]

A. Díaz, N. Harman, S. Howe and D. Thompson, Isoperimetric problems in sectors with density, Adv. Geom., 12 (2012), 589-619.   Google Scholar

[16]

J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z., 185 (1984), 339-353.  doi: 10.1007/BF01215045.  Google Scholar

[17]

A. Figalli and F. Maggi, On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations, 48 (2013), 447-489.  doi: 10.1007/s00526-012-0557-5.  Google Scholar

[18]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497.  doi: 10.1007/BF02566514.  Google Scholar

[19]

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.  doi: 10.1016/j.jfa.2011.01.007.  Google Scholar

[20]

L. Di Giosia, J. Habib, L. Kenigsberg, D. Pittman and W. Zhu, Balls Isoperimetric in ${\mathbb{R}}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$, preprint, arXiv: 1610.05830v1. Google Scholar

[21]

F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc., 355 (2003), 5041-5052.  doi: 10.1090/S0002-9947-03-03061-7.  Google Scholar

[22]

F. Morgan, Available from: http://sites.williams.edu/Morgan/2010/06/22/variation-formulae-for-perimeter-and-volume-densities/. Google Scholar

[23]

F. Morgan and A. Pratelli, Existence of isoperimetric regions in $\mathbb{R}^n$ with density, Ann. Global Anal. Geom., 43 (2013), 331-365.  doi: 10.1007/s10455-012-9348-7.  Google Scholar

[24]

W. Walter, Ordinary Differential Equations, English translation, Springer, 1998. doi: 10.1007/978-1-4612-0601-9.  Google Scholar

Figure 1.  Construction of $\Omega_i$ and $r_{i, j}$
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Figure 2.  the domain $\Omega_{\epsilon}$
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