\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Abstract Related Papers Cited by
  • 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 $p\in[1,d)$.
    Mathematics Subject Classification: Primary: 46E35, 26D10; Secondary: 52A38.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(363) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return