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

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

* 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.

Citation: Eric Baer, Alessio Figalli. Characterization of isoperimetric sets inside almost-convex cones. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press. , USA, 2000.  Google Scholar

[4]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Par. Diff. Eq., 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241.  Google Scholar

[5]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Applications 65, Birkhäuser, Boston, MA, 2005.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, Preprint, 2015, arXiv: 1211.3698. Google Scholar

[11]

R. Courant and D. Hilbert, Methods of Mathematical Physics, 1 (1953); 2 (1962). Wiley, New York. Google Scholar

[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. Google Scholar

[13]

G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, Switzerland, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. FigalliN. FuscoF. MaggiV. 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.  Google Scholar

[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.  Google Scholar

[20]

A. FigalliF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

N. FuscoF. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2), 168 (2008), 941-980.  doi: 10.4007/annals.2008.168.941.  Google Scholar

[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.  Google Scholar

[25]

R. HempelL. 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.  Google Scholar

[26]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Burkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

F. Morgan, Riemannian Geometry. A Beginner's Guide. 2nd ed. , A. K. Peters Ltd. , Wellesley, MA. , 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press. , USA, 2000.  Google Scholar

[4]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Comm. Par. Diff. Eq., 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241.  Google Scholar

[5]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Applications 65, Birkhäuser, Boston, MA, 2005.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, Preprint, 2015, arXiv: 1211.3698. Google Scholar

[11]

R. Courant and D. Hilbert, Methods of Mathematical Physics, 1 (1953); 2 (1962). Wiley, New York. Google Scholar

[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. Google Scholar

[13]

G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, Switzerland, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. FigalliN. FuscoF. MaggiV. 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.  Google Scholar

[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.  Google Scholar

[20]

A. FigalliF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

N. FuscoF. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2), 168 (2008), 941-980.  doi: 10.4007/annals.2008.168.941.  Google Scholar

[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.  Google Scholar

[25]

R. HempelL. 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.  Google Scholar

[26]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Burkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

F. Morgan, Riemannian Geometry. A Beginner's Guide. 2nd ed. , A. K. Peters Ltd. , Wellesley, MA. , 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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