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November
2018, 17(6): 2729-2749.
doi: 10.3934/cpaa.2018129
On the isoperimetric problem with perimeter density $r^p$
Universitat Politècnica de Catalunya, member of BGSMath, Barcelona, Spain |
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 and Applied Analysis,
2018, 17
(6)
: 2729-2749.
doi: 10.3934/cpaa.2018129
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
G. R. Chambers, Proof of the log-convex density conjecture, J. Eur. Math. Soc., to appear. |
| [11] |
G. Csató,
An isoperimetric problem with density and the Hardy-Sobolev inequality in ${\mathbb{R}}^2$, Differential Integral Equations, 28 (2015), 971-988.
|
| [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. |
| [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. |
| [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.
|
| [15] |
A. Díaz, N. Harman, S. Howe and D. Thompson,
Isoperimetric problems in sectors with density, Adv. Geom., 12 (2012), 589-619.
|
| [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. |
| [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. |
| [18] |
M. Flucher,
Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
| [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. |
| [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. |
| [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. |
| [22] |
F. Morgan, Available from: http://sites.williams.edu/Morgan/2010/06/22/variation-formulae-for-perimeter-and-volume-densities/. |
| [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. |
| [24] |
W. Walter, Ordinary Differential Equations, English translation, Springer, 1998.
doi: 10.1007/978-1-4612-0601-9. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
G. R. Chambers, Proof of the log-convex density conjecture, J. Eur. Math. Soc., to appear. |
| [11] |
G. Csató,
An isoperimetric problem with density and the Hardy-Sobolev inequality in ${\mathbb{R}}^2$, Differential Integral Equations, 28 (2015), 971-988.
|
| [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. |
| [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. |
| [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.
|
| [15] |
A. Díaz, N. Harman, S. Howe and D. Thompson,
Isoperimetric problems in sectors with density, Adv. Geom., 12 (2012), 589-619.
|
| [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. |
| [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. |
| [18] |
M. Flucher,
Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helvetici, 67 (1992), 471-497.
doi: 10.1007/BF02566514. |
| [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. |
| [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. |
| [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. |
| [22] |
F. Morgan, Available from: http://sites.williams.edu/Morgan/2010/06/22/variation-formulae-for-perimeter-and-volume-densities/. |
| [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. |
| [24] |
W. Walter, Ordinary Differential Equations, English translation, Springer, 1998.
doi: 10.1007/978-1-4612-0601-9. |
Figure 2.
the domain $\Omega_{\epsilon}$
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