Advanced Search
Article Contents
Article Contents

Characterization of isoperimetric sets inside almost-convex cones

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
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary:49Q20;Secondary:49K10.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   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.
      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.
      L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Univ. Press. , USA, 2000.
      A. Boulkhemair  and  A. Chakib , On the uniform Poincaré inequality, Comm. Par. Diff. Eq., 32 (2007) , 1439-1447.  doi: 10.1080/03605300600910241.
      D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Applications 65, Birkhäuser, Boston, MA, 2005.
      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.
      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.
      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.
      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.
      M. Cicalese, G. P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles, Preprint, 2015, arXiv: 1211.3698.
      R. Courant and D. Hilbert, Methods of Mathematical Physics, 1 (1953); 2 (1962). Wiley, New York.
      R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, 3, Transformations, Sobolev, Opérateurs, asson, Paris, 1984.
      G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, 233, Birkhäuser Verlag, Basel, Switzerland, 2005.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Burkhäuser Verlag, Basel, 2006.
      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.
      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.
      F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Univ. Press. , 2012. doi: 10.1017/CBO9781139108133.
      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.
      F. Morgan, Riemannian Geometry. A Beginner's Guide. 2nd ed. , A. K. Peters Ltd. , Wellesley, MA. , 1998.
      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.
      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.
      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.
      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.
  • 加载中
Open Access Under a Creative Commons license

Article Metrics

HTML views(1915) PDF downloads(262) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint