# American Institute of Mathematical Sciences

August  2013, 33(8): 3599-3640. doi: 10.3934/dcds.2013.33.3599

## Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities

 1 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy, Italy 2 Dipartimento di Matematica, Università di Roma "La Sapienza", Piazzale A. Moro 2, 00185 Roma, Italy

Received  August 2012 Revised  November 2012 Published  January 2013

We study weighted porous media equations on domains $\Omega\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $\Omega\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain $L^{q_0}$-$L^\varrho$ smoothing effects and suitable weighted Poincaré-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $\Omega={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted mean value instead than to zero. Examples are given in terms of wide classes of weights.
Citation: Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J., 30 (1981), 749-785. doi: 10.1512/iumj.1981.30.30056. [3] D. Andreucci, G. R. Cirmi, S. Leonardi and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174 (2001), 253-288. doi: 10.1006/jdeq.2000.3948. [4] D. Andreucci and A. F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Adv. Differential Equations, 5 (2000), 833-860. [5] D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Elect. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352. [6] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J., 44 (1995), 1033-1074. doi: 10.1512/iumj.1995.44.2019. [7] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436. doi: 10.1016/j.crma.2007.01.011. [8] M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107. [9] M. Bonforte and G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities, J. Funct. Anal., 225 (2005), 33-62. doi: 10.1016/j.jfa.2005.03.011. [10] M. Bonforte and G. Grillo, Ultracontractive bounds for nonlinear evolution equations governed by the subcritical $p$-Laplacian, in "Trends in Partial Differential Equations of Mathematical Physics," Progr. Nonlinear Differential Equations Appl., 61, Birkhäuser, Basel, (2005), 15-26. doi: 10.1007/3-7643-7317-2_2. [11] M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Equ., 8 (2008), 99-128. doi: 10.1007/s00028-007-0345-4. [12] M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemann manifold, Arch. Rat. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7. [13] S. M. Buckley and P. Koskela, New Poincaré inequalities from old, Ann. Acad. Sci. Fenn. Math., 23 (1998), 251-260. [14] S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities, Indiana Univ. Math. J., 49 (2000), 143-175. doi: 10.1512/iumj.2000.49.1754. [15] S.-K. Chua and R. L. Wheeden, Weighted Poincaré inequalities on convex domains, Math. Res. Lett., 17 (2010), 993-1011. [16] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. [17] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. [18] J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, $L^q$-functional inequalities and weighted porous media equations, Potential Anal., 28 (2008), 35-59. doi: 10.1007/s11118-007-9066-0. [19] J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., 6 (2008), 477-494. [20] D. E. Edmunds and B. Opic, Weighted Poincaré and Friedrichs inequalities, J. London Math. Soc. (2), 47 (1993), 79-96. doi: 10.1112/jlms/s2-47.1.79. [21] D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations, 84 (1990), 309-318. doi: 10.1016/0022-0396(90)90081-Y. [22] D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.2307/2160476. [23] E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. Röckner and D. W. Stroock, "Dirichlet Forms," Lectures given at the First C.I.M.E. Session held in Varenna, June 8-19, 1992, Edited by G. Dell'Antonio and U. Mosco, Lecture Notes in Mathematics, 1563, Springer-Verlag, Berlin, 1993. [24] P. Federbush, A partial alternate derivation of a result of Nelson, J. Math. Phys., 10 (1969), 50-52. [25] S. Filippas, L. Moschini and A. Tertikas, Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains, Comm. Math. Phys., 273 (2007), 237-281. doi: 10.1007/s00220-007-0253-z. [26] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions," Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6. [27] A. Grigor'yan, Heat kernels on weighted manifolds and applications, in "The Ubiquitous Heat Kernel," Contemp. Math., 398, Amer. Math. Soc., Providence, RI, (2006), 93-191. doi: 10.1090/conm/398/07486. [28] A. Grigor'yan, "Heat Kernel and Analysis on Manifolds," AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. [29] G. Grillo, On the equivalence between $p$-Poincaré inequalities and $L^r$-$L^q$ regularization and decay estimates of certain nonlinear evolutions, J. Differential Equations, 249 (2010), 2561-2576. doi: 10.1016/j.jde.2010.05.022. [30] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. [31] E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities," Courant Lecture Notes in Mathematics, 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [32] R. Hurri, The weighted Poincaré inequalities, Math. Scand., 67 (1990), 145-160. [33] S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521-549. doi: 10.3934/dcds.2010.26.521. [34] S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831-852. doi: 10.1002/cpa.3160340605. [35] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. doi: 10.1002/cpa.3160350106. [36] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537-554. [37] A. Kufner and B. Opic, "Hardy-Type Inequalities," Pitman Research Notes in Mathematics Series, 219, Longman Scientific & Technical, Harlow, 1990. [38] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. [39] G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [40] V. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. [41] B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38. [42] B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. [43] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [44] O. A. Oleĭnik, On the equations of unsteady filtration, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210-1213. [45] O. A. Oleĭnik, A. S. Kalašnikov and Y.-L. Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667-704. [46] M. M. Porzio, On decay estimates, J. Evol. Equ., 9 (2009), 561-591. doi: 10.1007/s00028-009-0024-8. [47] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. [48] G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337. [49] G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275. [50] G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508. doi: 10.3934/cpaa.2009.8.493. [51] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [52] J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001. [53] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [54] F.-Y. Wang, Orlicz-Poincaré inequalities, Proc. Edinb. Math. Soc. (2), 51 (2008), 529-543. doi: 10.1017/S0013091506000526.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. [2] N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J., 30 (1981), 749-785. doi: 10.1512/iumj.1981.30.30056. [3] D. Andreucci, G. R. Cirmi, S. Leonardi and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174 (2001), 253-288. doi: 10.1006/jdeq.2000.3948. [4] D. Andreucci and A. F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Adv. Differential Equations, 5 (2000), 833-860. [5] D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Elect. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352. [6] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J., 44 (1995), 1033-1074. doi: 10.1512/iumj.1995.44.2019. [7] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436. doi: 10.1016/j.crma.2007.01.011. [8] M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107. [9] M. Bonforte and G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities, J. Funct. Anal., 225 (2005), 33-62. doi: 10.1016/j.jfa.2005.03.011. [10] M. Bonforte and G. Grillo, Ultracontractive bounds for nonlinear evolution equations governed by the subcritical $p$-Laplacian, in "Trends in Partial Differential Equations of Mathematical Physics," Progr. Nonlinear Differential Equations Appl., 61, Birkhäuser, Basel, (2005), 15-26. doi: 10.1007/3-7643-7317-2_2. [11] M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Equ., 8 (2008), 99-128. doi: 10.1007/s00028-007-0345-4. [12] M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemann manifold, Arch. Rat. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7. [13] S. M. Buckley and P. Koskela, New Poincaré inequalities from old, Ann. Acad. Sci. Fenn. Math., 23 (1998), 251-260. [14] S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities, Indiana Univ. Math. J., 49 (2000), 143-175. doi: 10.1512/iumj.2000.49.1754. [15] S.-K. Chua and R. L. Wheeden, Weighted Poincaré inequalities on convex domains, Math. Res. Lett., 17 (2010), 993-1011. [16] E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. [17] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. [18] J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, $L^q$-functional inequalities and weighted porous media equations, Potential Anal., 28 (2008), 35-59. doi: 10.1007/s11118-007-9066-0. [19] J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., 6 (2008), 477-494. [20] D. E. Edmunds and B. Opic, Weighted Poincaré and Friedrichs inequalities, J. London Math. Soc. (2), 47 (1993), 79-96. doi: 10.1112/jlms/s2-47.1.79. [21] D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations, 84 (1990), 309-318. doi: 10.1016/0022-0396(90)90081-Y. [22] D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.2307/2160476. [23] E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. Röckner and D. W. Stroock, "Dirichlet Forms," Lectures given at the First C.I.M.E. Session held in Varenna, June 8-19, 1992, Edited by G. Dell'Antonio and U. Mosco, Lecture Notes in Mathematics, 1563, Springer-Verlag, Berlin, 1993. [24] P. Federbush, A partial alternate derivation of a result of Nelson, J. Math. Phys., 10 (1969), 50-52. [25] S. Filippas, L. Moschini and A. Tertikas, Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains, Comm. Math. Phys., 273 (2007), 237-281. doi: 10.1007/s00220-007-0253-z. [26] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions," Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6. [27] A. Grigor'yan, Heat kernels on weighted manifolds and applications, in "The Ubiquitous Heat Kernel," Contemp. Math., 398, Amer. Math. Soc., Providence, RI, (2006), 93-191. doi: 10.1090/conm/398/07486. [28] A. Grigor'yan, "Heat Kernel and Analysis on Manifolds," AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. [29] G. Grillo, On the equivalence between $p$-Poincaré inequalities and $L^r$-$L^q$ regularization and decay estimates of certain nonlinear evolutions, J. Differential Equations, 249 (2010), 2561-2576. doi: 10.1016/j.jde.2010.05.022. [30] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. [31] E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities," Courant Lecture Notes in Mathematics, 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. [32] R. Hurri, The weighted Poincaré inequalities, Math. Scand., 67 (1990), 145-160. [33] S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521-549. doi: 10.3934/dcds.2010.26.521. [34] S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831-852. doi: 10.1002/cpa.3160340605. [35] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. doi: 10.1002/cpa.3160350106. [36] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537-554. [37] A. Kufner and B. Opic, "Hardy-Type Inequalities," Pitman Research Notes in Mathematics Series, 219, Longman Scientific & Technical, Harlow, 1990. [38] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. [39] G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [40] V. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. [41] B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38. [42] B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. [43] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [44] O. A. Oleĭnik, On the equations of unsteady filtration, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210-1213. [45] O. A. Oleĭnik, A. S. Kalašnikov and Y.-L. Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667-704. [46] M. M. Porzio, On decay estimates, J. Evol. Equ., 9 (2009), 561-591. doi: 10.1007/s00028-009-0024-8. [47] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. [48] G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337. [49] G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275. [50] G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508. doi: 10.3934/cpaa.2009.8.493. [51] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [52] J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001. [53] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [54] F.-Y. Wang, Orlicz-Poincaré inequalities, Proc. Edinb. Math. Soc. (2), 51 (2008), 529-543. doi: 10.1017/S0013091506000526.
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