June  2015, 35(6): 2741-2761. doi: 10.3934/dcds.2015.35.2741

Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received  December 2013 Revised  April 2014 Published  December 2014

The aim of this paper is to prove an isoperimetric inequality relative to a convex domain $\Omega\subset\mathbb{R}^d$ intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius $r>0$ and the position $y\in\overline{\Omega}$ of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension $d,$ the diameter of the domain and the integrability exponent $p\in[1,d)$.
Citation: Marita Thomas. Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2741-2761. doi: 10.3934/dcds.2015.35.2741
References:
[1]

G. Acosta and R. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proceedings of the American Mathematical Society, 132 (2004), 195-202. doi: 10.1090/S0002-9939-03-07004-7.

[2]

R. Adams, Sobolev Spaces, Academic Press, 1975.

[3]

R. Adams and J. Fournier, Cone conditions and properties of Sobolev spaces, J. Math. Anal. Appl., 61 (1977), 713-734. doi: 10.1016/0022-247X(77)90173-1.

[4]

R. Adams and J. Fournier, Sobolev Spaces, 2nd edition, Elsevier, 2003.

[5]

S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., 1965.

[6]

S. Bartels, A. Mielke and T. Roubíček, Quasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal., 50 (2012), 951-976. doi: 10.1137/100819205.

[7]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commum. Part. Diff. Equat., 32 (2007), 1439-1447. doi: 10.1080/03605300600910241.

[8]

Y. Burago and V. Zalgaller, Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften, 285. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[10]

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.

[11]

D. Chenais, Sur une famille de variétés a bord lipschitziennes. application à un problème d' identification de domaines, Annales de'l Institut Fourier, 27 (1977), 201-231. doi: 10.5802/aif.676.

[12]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, A temperature-dependent phase segregation problem of the Allen-Cahn type, Adv. Math. Sci. Appl., 20 (2010), 219-234.

[13]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870. doi: 10.1137/110828526.

[14]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital., 5 (2012), 495-513.

[15]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368.

[16]

G. Colombo and K. Nguyen, Quantitative isoperimetric inequalities for a class of nonconvex sets, Calc. Var., 37 (2010), 141-166. doi: 10.1007/s00526-009-0256-z.

[17]

G. Dal Maso, A. DeSimone and M. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291. doi: 10.1007/s00205-005-0407-0.

[18]

G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.

[19]

A. DeSimone and M. Kružík, Domain patterns and hysteresis in phase-transforming solids: analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation, Networks and Heterogeneous Media, 8 (2013), 481-499. doi: 10.3934/nhm.2013.8.481.

[20]

W. Dreyer and C. Guhlke, Sharp limit of the viscous Cahn-Hilliard equation and thermodynamic consistency, WIAS-Preprint 1771.

[21]

L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851. doi: 10.1007/s00205-012-0545-0.

[22]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.

[23]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[24]

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.

[25]

G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium, Eur. J. Mech., A/Solids, 12 (1993), 149-189.

[26]

S. Frigeri, M. Grasselli and Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differential Equations, 255 (2013), 2587-2614. doi: 10.1016/j.jde.2013.07.016.

[27]

B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $\mathbbR^n$, Transactions of the American Mathematical Society, 314 (1989), 619-638. doi: 10.2307/2001401.

[28]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, Journal of Functional Analysis, 244 (2007), 315-341. doi: 10.1016/j.jfa.2006.10.015.

[29]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Annals of Mathematics, 168 (2008), 941-980. doi: 10.4007/annals.2008.168.941.

[30]

H. Gajewski, An application of eigenfunctions of $p$-Laplacians to domain separation, Mathematica Bohemica, 126 (2001), 395-401.

[31]

H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of $p$-Laplacians, Appl. Anal., 79 (2001), 483-501. doi: 10.1080/00036810108840974.

[32]

H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of $p$-Laplacians, II, Nonlinear Anal., 52 (2003), 291-304.

[33]

H. Garcke and C. Kraus, An inhomogeneous, anisotropic, elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, AMSA, 20 (2010), 511-545.

[34]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture, Calc. Var. Partial Differential Equations, 22 (2005), 129-172. doi: 10.1007/s00526-004-0269-6.

[35]

J. Griepentrog, K. Gröger, H.-K. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Mathematische Nachrichten, 241 (2002), 110-120. doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R.

[36]

J. Griepentrog and L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on sobolev-campanato spaces, Mathematische Nachrichten, 225 (2001), 39-74. doi: 10.1002/1522-2616(200105)225:1<39::AID-MANA39>3.0.CO;2-5.

[37]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.

[38]

K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687. doi: 10.1007/BF01442860.

[39]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, J. Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[40]

F. Hausdorff, Set Theory, Translated from the German by John R. Aumann et al Chelsea Publishing Co., New York, 1962.

[41]

I. Ly and D. Seck, Isoperimetric inequality for an interior free boundary problem with $p$-Laplacian operator, EJDE, 2004 (2004), 1-12.

[42]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, Journal of Geometric Analysis, 15 (2005), 83-121. doi: 10.1007/BF02921860.

[43]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities. Part II: Variants and extensions, Calc. Var., 31 (2008), 47-74. doi: 10.1007/s00526-007-0105-x.

[44]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-15564-2.

[45]

L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains, Archive for Rational Mechanics and Analysis, 5 (1860), 286-292. doi: 10.1007/BF00252910.

[46]

W. Pfeffer, Derivation and Integration, Cambridge University Press, 2001. doi: 10.1017/CBO9780511574764.

[47]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, 1984. doi: 10.1007/978-3-642-87722-3.

[48]

J. Rehberg, A criterion for a two-dimensional domain to be Lipschitzian, WIAS-Preprint, 1695.

[49]

R. Rockafellar and R.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3.

[50]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, WIAS-Preprint 1692, E-First in ESAIM-COCV, 2014.

[51]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle, delamination, WIAS-Preprint 1889, to appear in Nonlinear Anal. Real World Appl., 2015.

[52]

G. Schimperna and U. Stefanelli, Positivity of the temperature for phase transitions with micro-movements, Nonlinear Anal. Real World Appl., 8 (2007), 257-266. doi: 10.1016/j.nonrwa.2005.08.004.

[53]

P.-M. Suquet, Existence et régularité des solutions des équations de la plasticité, C. R. Acad. Sci. Paris Sér. A-B, 286 (1978), A1201-A1204.

[54]

W. Ziemer, Weakly Differentiable Functions, Springer, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1]

G. Acosta and R. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proceedings of the American Mathematical Society, 132 (2004), 195-202. doi: 10.1090/S0002-9939-03-07004-7.

[2]

R. Adams, Sobolev Spaces, Academic Press, 1975.

[3]

R. Adams and J. Fournier, Cone conditions and properties of Sobolev spaces, J. Math. Anal. Appl., 61 (1977), 713-734. doi: 10.1016/0022-247X(77)90173-1.

[4]

R. Adams and J. Fournier, Sobolev Spaces, 2nd edition, Elsevier, 2003.

[5]

S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., 1965.

[6]

S. Bartels, A. Mielke and T. Roubíček, Quasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal., 50 (2012), 951-976. doi: 10.1137/100819205.

[7]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commum. Part. Diff. Equat., 32 (2007), 1439-1447. doi: 10.1080/03605300600910241.

[8]

Y. Burago and V. Zalgaller, Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften, 285. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-662-07441-1.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[10]

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.

[11]

D. Chenais, Sur une famille de variétés a bord lipschitziennes. application à un problème d' identification de domaines, Annales de'l Institut Fourier, 27 (1977), 201-231. doi: 10.5802/aif.676.

[12]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, A temperature-dependent phase segregation problem of the Allen-Cahn type, Adv. Math. Sci. Appl., 20 (2010), 219-234.

[13]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870. doi: 10.1137/110828526.

[14]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital., 5 (2012), 495-513.

[15]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368.

[16]

G. Colombo and K. Nguyen, Quantitative isoperimetric inequalities for a class of nonconvex sets, Calc. Var., 37 (2010), 141-166. doi: 10.1007/s00526-009-0256-z.

[17]

G. Dal Maso, A. DeSimone and M. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291. doi: 10.1007/s00205-005-0407-0.

[18]

G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.

[19]

A. DeSimone and M. Kružík, Domain patterns and hysteresis in phase-transforming solids: analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation, Networks and Heterogeneous Media, 8 (2013), 481-499. doi: 10.3934/nhm.2013.8.481.

[20]

W. Dreyer and C. Guhlke, Sharp limit of the viscous Cahn-Hilliard equation and thermodynamic consistency, WIAS-Preprint 1771.

[21]

L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851. doi: 10.1007/s00205-012-0545-0.

[22]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.

[23]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[24]

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.

[25]

G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium, Eur. J. Mech., A/Solids, 12 (1993), 149-189.

[26]

S. Frigeri, M. Grasselli and Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differential Equations, 255 (2013), 2587-2614. doi: 10.1016/j.jde.2013.07.016.

[27]

B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $\mathbbR^n$, Transactions of the American Mathematical Society, 314 (1989), 619-638. doi: 10.2307/2001401.

[28]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, Journal of Functional Analysis, 244 (2007), 315-341. doi: 10.1016/j.jfa.2006.10.015.

[29]

N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Annals of Mathematics, 168 (2008), 941-980. doi: 10.4007/annals.2008.168.941.

[30]

H. Gajewski, An application of eigenfunctions of $p$-Laplacians to domain separation, Mathematica Bohemica, 126 (2001), 395-401.

[31]

H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of $p$-Laplacians, Appl. Anal., 79 (2001), 483-501. doi: 10.1080/00036810108840974.

[32]

H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of $p$-Laplacians, II, Nonlinear Anal., 52 (2003), 291-304.

[33]

H. Garcke and C. Kraus, An inhomogeneous, anisotropic, elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, AMSA, 20 (2010), 511-545.

[34]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture, Calc. Var. Partial Differential Equations, 22 (2005), 129-172. doi: 10.1007/s00526-004-0269-6.

[35]

J. Griepentrog, K. Gröger, H.-K. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Mathematische Nachrichten, 241 (2002), 110-120. doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R.

[36]

J. Griepentrog and L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on sobolev-campanato spaces, Mathematische Nachrichten, 225 (2001), 39-74. doi: 10.1002/1522-2616(200105)225:1<39::AID-MANA39>3.0.CO;2-5.

[37]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.

[38]

K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687. doi: 10.1007/BF01442860.

[39]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, J. Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[40]

F. Hausdorff, Set Theory, Translated from the German by John R. Aumann et al Chelsea Publishing Co., New York, 1962.

[41]

I. Ly and D. Seck, Isoperimetric inequality for an interior free boundary problem with $p$-Laplacian operator, EJDE, 2004 (2004), 1-12.

[42]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, Journal of Geometric Analysis, 15 (2005), 83-121. doi: 10.1007/BF02921860.

[43]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities. Part II: Variants and extensions, Calc. Var., 31 (2008), 47-74. doi: 10.1007/s00526-007-0105-x.

[44]

V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, 2011. doi: 10.1007/978-3-642-15564-2.

[45]

L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains, Archive for Rational Mechanics and Analysis, 5 (1860), 286-292. doi: 10.1007/BF00252910.

[46]

W. Pfeffer, Derivation and Integration, Cambridge University Press, 2001. doi: 10.1017/CBO9780511574764.

[47]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, 1984. doi: 10.1007/978-3-642-87722-3.

[48]

J. Rehberg, A criterion for a two-dimensional domain to be Lipschitzian, WIAS-Preprint, 1695.

[49]

R. Rockafellar and R.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3.

[50]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, WIAS-Preprint 1692, E-First in ESAIM-COCV, 2014.

[51]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle, delamination, WIAS-Preprint 1889, to appear in Nonlinear Anal. Real World Appl., 2015.

[52]

G. Schimperna and U. Stefanelli, Positivity of the temperature for phase transitions with micro-movements, Nonlinear Anal. Real World Appl., 8 (2007), 257-266. doi: 10.1016/j.nonrwa.2005.08.004.

[53]

P.-M. Suquet, Existence et régularité des solutions des équations de la plasticité, C. R. Acad. Sci. Paris Sér. A-B, 286 (1978), A1201-A1204.

[54]

W. Ziemer, Weakly Differentiable Functions, Springer, 1989. doi: 10.1007/978-1-4612-1015-3.

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