American Institute of Mathematical Sciences

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

Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Marita Thomas 1,

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 $p\in[1,d)$.
Keywords: uniform cone property., Poincaré-Sobolev inequality, relative isoperimetric inequality.
Mathematics Subject Classification: Primary: 46E35, 26D10; Secondary: 52A3.
Citation: Marita Thomas. Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1090/S0002-9939-03-07004-7. Google Scholar

[2]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[3]

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

[4]

R. Adams and J. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

[5]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (1965). Google Scholar

[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. doi: 10.1137/100819205. Google Scholar

[7]

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

[8]

Y. Burago and V. Zalgaller, Geometric Inequalities,, Grundlehren der Mathematischen Wissenschaften, (1988). doi: 10.1007/978-3-662-07441-1. Google Scholar

[9]

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

[10]

D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[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. doi: 10.5802/aif.676. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/110828526. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. doi: 10.1007/s00205-005-0407-0. Google Scholar

[18]

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

[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. doi: 10.3934/nhm.2013.8.481. Google Scholar

[20]

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

[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. doi: 10.1007/s00205-012-0545-0. Google Scholar

[22]

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

[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. doi: 10.1002/mma.1089. Google Scholar

[24]

A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities,, Invent math., 182 (2010), 167. doi: 10.1007/s00222-010-0261-z. Google Scholar

[25]

G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium,, Eur. J. Mech., 12 (1993), 149. Google Scholar

[26]

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

[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. doi: 10.2307/2001401. Google Scholar

[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. doi: 10.1016/j.jfa.2006.10.015. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[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. Google Scholar

[34]

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

[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. doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R. Google Scholar

[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. doi: 10.1002/1522-2616(200105)225:1<39::AID-MANA39>3.0.CO;2-5. Google Scholar

[37]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). Google Scholar

[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. doi: 10.1007/BF01442860. Google Scholar

[39]

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

[40]

F. Hausdorff, Set Theory,, Translated from the German by John R. Aumann et al Chelsea Publishing Co., (1962). Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

[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). Google Scholar

[51]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle,, delamination, (1889). Google Scholar

[52]

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

[53]

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

[54]

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

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. doi: 10.1090/S0002-9939-03-07004-7. Google Scholar

[2]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[3]

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

[4]

R. Adams and J. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

[5]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, D. Van Nostrand Company, (1965). Google Scholar

[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. doi: 10.1137/100819205. Google Scholar

[7]

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

[8]

Y. Burago and V. Zalgaller, Geometric Inequalities,, Grundlehren der Mathematischen Wissenschaften, (1988). doi: 10.1007/978-3-662-07441-1. Google Scholar

[9]

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

[10]

D. Chenais, On the existence of a solution in a domain identification problem,, J. Math. Anal. Appl., 52 (1975), 189. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[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. doi: 10.5802/aif.676. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/110828526. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. doi: 10.1007/s00205-005-0407-0. Google Scholar

[18]

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

[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. doi: 10.3934/nhm.2013.8.481. Google Scholar

[20]

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

[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. doi: 10.1007/s00205-012-0545-0. Google Scholar

[22]

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

[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. doi: 10.1002/mma.1089. Google Scholar

[24]

A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities,, Invent math., 182 (2010), 167. doi: 10.1007/s00222-010-0261-z. Google Scholar

[25]

G. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium,, Eur. J. Mech., 12 (1993), 149. Google Scholar

[26]

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

[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. doi: 10.2307/2001401. Google Scholar

[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. doi: 10.1016/j.jfa.2006.10.015. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[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. Google Scholar

[34]

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

[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. doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R. Google Scholar

[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. doi: 10.1002/1522-2616(200105)225:1<39::AID-MANA39>3.0.CO;2-5. Google Scholar

[37]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). Google Scholar

[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. doi: 10.1007/BF01442860. Google Scholar

[39]

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

[40]

F. Hausdorff, Set Theory,, Translated from the German by John R. Aumann et al Chelsea Publishing Co., (1962). Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

[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). Google Scholar

[51]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle,, delamination, (1889). Google Scholar

[52]

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

[53]

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

[54]

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

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