Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (6) : 2741-2761. doi: 10.3934/dcds.2015.35.2741

References:

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[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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