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January  2017, 37(1): 1-14. doi: 10.3934/dcds.2017001

Characterization of isoperimetric sets inside almost-convex cones

Eric Baer 1, and  Alessio Figalli 2,,

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

* Corresponding author:figalli@math.utexas.edu

Received  April 2016 Revised  May 2016 Published  November 2016

Fund Project: The work of E.B. was partially supported by the National Science Foundation under Award Nos. DMS-1204557 and DMS-1147523. The work of A.F. was partially supported by the National Science Foundation under Award Nos. DMS-1262411 and DMS-1361122.

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.

Keywords: Isoperimetric inequality, almost-convex cones.
Mathematics Subject Classification: Primary:49Q20;Secondary:49K1.
Citation: Eric Baer, Alessio Figalli. Characterization of isoperimetric sets inside almost-convex cones. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 1-14. doi: 10.3934/dcds.2017001
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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