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

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

References:

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[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
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[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
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[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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