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Characterization of isoperimetric sets inside almost-convex cones
1. | Department of Mathematics, University of Wisconsin Madison, 480 Lincoln Dr, Madison, WI 53706, USA |
2. | The University of Texas at Austin, Mathematics Department, 2515 Speedway Stop C1200, Austin, TX 78712, USA |
In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersecting the cone with a ball centered at the origin.
References:
[1] |
Y. Alkhutov and V. G. Maz'ya, L1, p-coercitivity and estimates of the Green function of the Neumann problem in a convex domain, J. Math. Sci. , New York, 196 (2014), 245-261; English translation of Probl. Mat. Anal. , 73 (2013), 3-16 (Russian).
doi: 10.1007/s10958-014-1656-y. |
[2] |
F. J. Almgren, Jr. , Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), ⅷ+199 pp. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press. , USA, 2000. |
[4] |
A. Boulkhemair and A. Chakib,
On the uniform Poincaré inequality, Comm. Par. Diff. Eq., 32 (2007), 1439-1447.
doi: 10.1080/03605300600910241. |
[5] |
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Applications 65, Birkhäuser, Boston, MA, 2005. |
[6] |
X. Cabré, X. Ros-Oton and J. Serra, Sharp isoperimetric inequalities via the ABP method, To appear in J. Eur. Math. Soc. , arXiv: 1304.1724. |
[7] |
A. Cañete and C. Rosales,
Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, Calc. Var. Par. Diff. Eq., 51 (2014), 887-913.
doi: 10.1007/s00526-013-0699-0. |
[8] |
D. Chenais,
On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219.
doi: 10.1016/0022-247X(75)90091-8. |
[9] |
M. Cicalese and G.P. Leonardi,
A selection principle for the sharp quantitative isoperimetric
inequality, Arch. Ration. Mech. Anal., 206 (2012), 617-643.
doi: 10.1007/s00205-012-0544-1. |
[10] |
M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, Preprint, 2015, arXiv: 1211.3698. |
[11] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, 1 (1953); 2 (1962). Wiley, New York. |
[12] |
R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, 3, Transformations, Sobolev, Opérateurs, asson, Paris, 1984. |
[13] |
G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, Switzerland, 2005. |
[14] |
G. De Philippis and F. Maggi,
Regularity of free boundaries in anisotropic capillarity problems
and the validity of Young's law, Arch. Ration. Mech. Anal., 216 (2015), 473-568.
doi: 10.1007/s00205-014-0813-2. |
[15] |
E. Durand-Cartagena and A. Lemenant,
Some stability results under domain variation for
Neumann problems in metric spaces, Ann. Acad. Sci. Fenn. Math., 35 (2010), 537-563.
doi: 10.5186/aasfm.2010.3533. |
[16] |
J.F. Escobar,
Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities
and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857-883.
doi: 10.1002/cpa.3160430703. |
[17] |
A. Figalli and E. Indrei,
A sharp stability result for the relative isoperimetric inequality inside
convex cones, J. Geom. Anal., 23 (2013), 938-969.
doi: 10.1007/s12220-011-9270-4. |
[18] |
A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini,
Isoperimetry and stability properties
of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.
doi: 10.1007/s00220-014-2244-1. |
[19] |
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. |
[20] |
A. Figalli, F. Maggi and A. Pratelli,
A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211.
doi: 10.1007/s00222-010-0261-z. |
[21] |
B. Fuglede,
Stability in the isoperimetric problem for convex or nearly spherical domains in ℝn, Trans. Amer. Math. Soc., 314 (1989), 619-638.
doi: 10.2307/2001401. |
[22] |
N. Fusco and V. Julin,
A strong form of the quantitative isoperimetric inequality, Calc. Var. Partial Differential Equations, 50 (2014), 925-937.
doi: 10.1007/s00526-013-0661-1. |
[23] |
N. Fusco, F. Maggi and A. Pratelli,
The sharp quantitative isoperimetric inequality, Ann. of Math. (2), 168 (2008), 941-980.
doi: 10.4007/annals.2008.168.941. |
[24] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Math. , 80. Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[25] |
R. Hempel, L. Seco and B. Simon,
The essential spectrum of Neumann Laplacians on some
bounded singular domain, J. Funct. Anal., 102 (1991), 448-483.
doi: 10.1016/0022-1236(91)90130-W. |
[26] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Burkhäuser Verlag, Basel, 2006. |
[27] |
P.-L. Lions and F. Pacella,
Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485.
doi: 10.1090/S0002-9939-1990-1000160-1. |
[28] |
J. -L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅰ. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, New York-Heidelberg, 1972. |
[29] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Univ. Press. , 2012.
doi: 10.1017/CBO9781139108133. |
[30] |
V. G. Maz'ya, On the boundedness of first derivatives for solutions to the Neumann-Laplace problem in a convex domain, J. Math. Sci. , New York, 159 (2009), 104-112; English translation of Probl. Mat. Anal. , 40 (2009), 105-112. (Russian).
doi: 10.1007/s10958-009-9430-2. |
[31] |
F. Morgan, Riemannian Geometry. A Beginner's Guide. 2nd ed. , A. K. Peters Ltd. , Wellesley, MA. , 1998. |
[32] |
F. Morgan and M. Ritoré,
Isoperimetric regions in cones, Trans. Amer. Math. Soc., 354 (2002), 2327-2339.
doi: 10.1090/S0002-9947-02-02983-5. |
[33] |
M. Ritoré and C. Rosales,
Existence and characterization of regions minimizing perimeter
under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc., 356 (2004), 4601-4622.
doi: 10.1090/S0002-9947-04-03537-8. |
[34] |
M. Ritoré and E. Vernadakis,
Isoperimetric inequalities in convex cylinders and cylindrically
bounded convex bodies, Calc. Var. Par. Diff. Eq., 54 (2015), 643-663.
doi: 10.1007/s00526-014-0800-3. |
[35] |
D. Ruiz,
On the uniformity of the constant in the Poincaré inequality, Adv. Nonlinear Stud., 12 (2012), 889-903.
doi: 10.1515/ans-2012-0413. |
show all references
References:
[1] |
Y. Alkhutov and V. G. Maz'ya, L1, p-coercitivity and estimates of the Green function of the Neumann problem in a convex domain, J. Math. Sci. , New York, 196 (2014), 245-261; English translation of Probl. Mat. Anal. , 73 (2013), 3-16 (Russian).
doi: 10.1007/s10958-014-1656-y. |
[2] |
F. J. Almgren, Jr. , Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), ⅷ+199 pp. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press. , USA, 2000. |
[4] |
A. Boulkhemair and A. Chakib,
On the uniform Poincaré inequality, Comm. Par. Diff. Eq., 32 (2007), 1439-1447.
doi: 10.1080/03605300600910241. |
[5] |
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Applications 65, Birkhäuser, Boston, MA, 2005. |
[6] |
X. Cabré, X. Ros-Oton and J. Serra, Sharp isoperimetric inequalities via the ABP method, To appear in J. Eur. Math. Soc. , arXiv: 1304.1724. |
[7] |
A. Cañete and C. Rosales,
Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities, Calc. Var. Par. Diff. Eq., 51 (2014), 887-913.
doi: 10.1007/s00526-013-0699-0. |
[8] |
D. Chenais,
On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219.
doi: 10.1016/0022-247X(75)90091-8. |
[9] |
M. Cicalese and G.P. Leonardi,
A selection principle for the sharp quantitative isoperimetric
inequality, Arch. Ration. Mech. Anal., 206 (2012), 617-643.
doi: 10.1007/s00205-012-0544-1. |
[10] |
M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, Preprint, 2015, arXiv: 1211.3698. |
[11] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, 1 (1953); 2 (1962). Wiley, New York. |
[12] |
R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, 3, Transformations, Sobolev, Opérateurs, asson, Paris, 1984. |
[13] |
G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, Switzerland, 2005. |
[14] |
G. De Philippis and F. Maggi,
Regularity of free boundaries in anisotropic capillarity problems
and the validity of Young's law, Arch. Ration. Mech. Anal., 216 (2015), 473-568.
doi: 10.1007/s00205-014-0813-2. |
[15] |
E. Durand-Cartagena and A. Lemenant,
Some stability results under domain variation for
Neumann problems in metric spaces, Ann. Acad. Sci. Fenn. Math., 35 (2010), 537-563.
doi: 10.5186/aasfm.2010.3533. |
[16] |
J.F. Escobar,
Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities
and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857-883.
doi: 10.1002/cpa.3160430703. |
[17] |
A. Figalli and E. Indrei,
A sharp stability result for the relative isoperimetric inequality inside
convex cones, J. Geom. Anal., 23 (2013), 938-969.
doi: 10.1007/s12220-011-9270-4. |
[18] |
A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini,
Isoperimetry and stability properties
of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.
doi: 10.1007/s00220-014-2244-1. |
[19] |
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. |
[20] |
A. Figalli, F. Maggi and A. Pratelli,
A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211.
doi: 10.1007/s00222-010-0261-z. |
[21] |
B. Fuglede,
Stability in the isoperimetric problem for convex or nearly spherical domains in ℝn, Trans. Amer. Math. Soc., 314 (1989), 619-638.
doi: 10.2307/2001401. |
[22] |
N. Fusco and V. Julin,
A strong form of the quantitative isoperimetric inequality, Calc. Var. Partial Differential Equations, 50 (2014), 925-937.
doi: 10.1007/s00526-013-0661-1. |
[23] |
N. Fusco, F. Maggi and A. Pratelli,
The sharp quantitative isoperimetric inequality, Ann. of Math. (2), 168 (2008), 941-980.
doi: 10.4007/annals.2008.168.941. |
[24] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Math. , 80. Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[25] |
R. Hempel, L. Seco and B. Simon,
The essential spectrum of Neumann Laplacians on some
bounded singular domain, J. Funct. Anal., 102 (1991), 448-483.
doi: 10.1016/0022-1236(91)90130-W. |
[26] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Burkhäuser Verlag, Basel, 2006. |
[27] |
P.-L. Lions and F. Pacella,
Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485.
doi: 10.1090/S0002-9939-1990-1000160-1. |
[28] |
J. -L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. Ⅰ. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, New York-Heidelberg, 1972. |
[29] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Univ. Press. , 2012.
doi: 10.1017/CBO9781139108133. |
[30] |
V. G. Maz'ya, On the boundedness of first derivatives for solutions to the Neumann-Laplace problem in a convex domain, J. Math. Sci. , New York, 159 (2009), 104-112; English translation of Probl. Mat. Anal. , 40 (2009), 105-112. (Russian).
doi: 10.1007/s10958-009-9430-2. |
[31] |
F. Morgan, Riemannian Geometry. A Beginner's Guide. 2nd ed. , A. K. Peters Ltd. , Wellesley, MA. , 1998. |
[32] |
F. Morgan and M. Ritoré,
Isoperimetric regions in cones, Trans. Amer. Math. Soc., 354 (2002), 2327-2339.
doi: 10.1090/S0002-9947-02-02983-5. |
[33] |
M. Ritoré and C. Rosales,
Existence and characterization of regions minimizing perimeter
under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc., 356 (2004), 4601-4622.
doi: 10.1090/S0002-9947-04-03537-8. |
[34] |
M. Ritoré and E. Vernadakis,
Isoperimetric inequalities in convex cylinders and cylindrically
bounded convex bodies, Calc. Var. Par. Diff. Eq., 54 (2015), 643-663.
doi: 10.1007/s00526-014-0800-3. |
[35] |
D. Ruiz,
On the uniformity of the constant in the Poincaré inequality, Adv. Nonlinear Stud., 12 (2012), 889-903.
doi: 10.1515/ans-2012-0413. |
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