October  2015, 35(10): 4791-4804. doi: 10.3934/dcds.2015.35.4791

Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, South Korea

2. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, South Korea

Received  August 2014 Revised  February 2015 Published  April 2015

A global Calderón-Zygmund estimate type estimate in Weighted Lorentz spaces and Lorentz-Morrey spaces is obtained for weak solutions to elliptic obstacle problems of $p$-Laplacian type with discontinuous coefficients over Reifenberg flat domains.
Citation: Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4791-4804. doi: 10.3934/dcds.2015.35.4791
References:
[1]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188. doi: 10.1016/j.na.2013.11.004.  Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[3]

I. Blank and Z. Hao, Reifenberg Flatness of Free Boundaries in Obstacle Problems with VMO Ingredients, Calc. Var. Partial Differential Equations, 2014. doi: 10.1007/s00526-014-0772-3.  Google Scholar

[4]

V. Bögelein, F. Duzaar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160. doi: 10.1515/CRELLE.2011.006.  Google Scholar

[5]

S. Byun and D. K. Palagachev, Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains, Calc. Var. Partial Differential Equations, 49 (2014), 37-76. doi: 10.1007/s00526-012-0574-4.  Google Scholar

[6]

S. Byun, D. Palagachev and S. Ryu, Weighted $W^{1,p}$ estimates for solutions of non-linear parabolic equations over non-smooth domains, Bull. London Math. Soc., 45 (2013), 765-778. doi: 10.1112/blms/bdt011.  Google Scholar

[7]

S. Byun, Y. Cho and D. Palagachev, Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains, Proc. Amer. Math. Soc., (2015). doi: 10.1090/S0002-9939-2015-12458-6.  Google Scholar

[8]

S. Byun, Y. Cho and L. Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal., 263 (2012), 3117-3143. doi: 10.1016/j.jfa.2012.07.018.  Google Scholar

[9]

M. Eleuteri and J. Habermann, Calderón-Zygmund type estimates for a class of obstacle problems with $p(x)$-growth, J. Math. Anal. Appl., 372 (2010), 140-161. doi: 10.1016/j.jmaa.2010.05.072.  Google Scholar

[10]

A. Erhardt, Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth, Adv. Nonlinear Anal., 3 (2014), 15-44. doi: 10.1515/anona-2013-0024.  Google Scholar

[11]

A. Friedman, Variational Principles and Free-Boundary Problems, Second edition, Robert E. Krieger Publishing Co. Inc., Malabar, FL, 1988.  Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics, 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.  Google Scholar

[13]

M. Gupta and A. Bhar, On Lorentz and Orlicz-Lorentz subspaces of bounded families and approximation type operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 108 (2014), 733-755. doi: 10.1007/s13398-013-0137-3.  Google Scholar

[14]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[15]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.  Google Scholar

[16]

H. Kempka and J. Vybíral, Lorentz spaces with variable exponents, Math. Nachr., 287 (2014), 938-954. doi: 10.1002/mana.201200278.  Google Scholar

[17]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar

[18]

G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55. doi: 10.2307/1969496.  Google Scholar

[19]

T. Mengesha and N. C. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250 (2011), 2485-2507. doi: 10.1016/j.jde.2010.11.009.  Google Scholar

[20]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal., 203 (2012), 189-216. doi: 10.1007/s00205-011-0446-7.  Google Scholar

[21]

T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework,, , ().   Google Scholar

[22]

N. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189-206.  Google Scholar

[23]

G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 6 (2007), 195-261.  Google Scholar

[24]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[25]

N. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math., 250 (2014), 387-419. doi: 10.1016/j.aim.2013.09.022.  Google Scholar

[26]

E. Reifenberg, Solutions of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92. doi: 10.1007/BF02547186.  Google Scholar

[27]

J. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, Notas de Matemática [Mathematical Notes], 114, North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[28]

C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal., 262 (2012), 2777-2832. doi: 10.1016/j.jfa.2012.01.003.  Google Scholar

[29]

C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Manuscripta Math., 146 (2015), 7-63. doi: 10.1007/s00229-014-0684-8.  Google Scholar

[30]

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971.  Google Scholar

[31]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.  Google Scholar

[32]

F. Yao, Global higher integrability for the parabolic equations in Reifenberg domains, Math. Nachr., 283 (2010), 1358-1367. doi: 10.1002/mana.200710084.  Google Scholar

[33]

C. Zhang and S. Zhou, Global weighted estimates for quasilinear elliptic equations with non-standard growth, J. Funct. Anal., 267 (2014), 605-642. doi: 10.1016/j.jfa.2014.03.022.  Google Scholar

show all references

References:
[1]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188. doi: 10.1016/j.na.2013.11.004.  Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[3]

I. Blank and Z. Hao, Reifenberg Flatness of Free Boundaries in Obstacle Problems with VMO Ingredients, Calc. Var. Partial Differential Equations, 2014. doi: 10.1007/s00526-014-0772-3.  Google Scholar

[4]

V. Bögelein, F. Duzaar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160. doi: 10.1515/CRELLE.2011.006.  Google Scholar

[5]

S. Byun and D. K. Palagachev, Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains, Calc. Var. Partial Differential Equations, 49 (2014), 37-76. doi: 10.1007/s00526-012-0574-4.  Google Scholar

[6]

S. Byun, D. Palagachev and S. Ryu, Weighted $W^{1,p}$ estimates for solutions of non-linear parabolic equations over non-smooth domains, Bull. London Math. Soc., 45 (2013), 765-778. doi: 10.1112/blms/bdt011.  Google Scholar

[7]

S. Byun, Y. Cho and D. Palagachev, Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains, Proc. Amer. Math. Soc., (2015). doi: 10.1090/S0002-9939-2015-12458-6.  Google Scholar

[8]

S. Byun, Y. Cho and L. Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal., 263 (2012), 3117-3143. doi: 10.1016/j.jfa.2012.07.018.  Google Scholar

[9]

M. Eleuteri and J. Habermann, Calderón-Zygmund type estimates for a class of obstacle problems with $p(x)$-growth, J. Math. Anal. Appl., 372 (2010), 140-161. doi: 10.1016/j.jmaa.2010.05.072.  Google Scholar

[10]

A. Erhardt, Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth, Adv. Nonlinear Anal., 3 (2014), 15-44. doi: 10.1515/anona-2013-0024.  Google Scholar

[11]

A. Friedman, Variational Principles and Free-Boundary Problems, Second edition, Robert E. Krieger Publishing Co. Inc., Malabar, FL, 1988.  Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics, 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.  Google Scholar

[13]

M. Gupta and A. Bhar, On Lorentz and Orlicz-Lorentz subspaces of bounded families and approximation type operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 108 (2014), 733-755. doi: 10.1007/s13398-013-0137-3.  Google Scholar

[14]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.  Google Scholar

[15]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.  Google Scholar

[16]

H. Kempka and J. Vybíral, Lorentz spaces with variable exponents, Math. Nachr., 287 (2014), 938-954. doi: 10.1002/mana.201200278.  Google Scholar

[17]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar

[18]

G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55. doi: 10.2307/1969496.  Google Scholar

[19]

T. Mengesha and N. C. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250 (2011), 2485-2507. doi: 10.1016/j.jde.2010.11.009.  Google Scholar

[20]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal., 203 (2012), 189-216. doi: 10.1007/s00205-011-0446-7.  Google Scholar

[21]

T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework,, , ().   Google Scholar

[22]

N. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189-206.  Google Scholar

[23]

G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 6 (2007), 195-261.  Google Scholar

[24]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[25]

N. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math., 250 (2014), 387-419. doi: 10.1016/j.aim.2013.09.022.  Google Scholar

[26]

E. Reifenberg, Solutions of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92. doi: 10.1007/BF02547186.  Google Scholar

[27]

J. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, Notas de Matemática [Mathematical Notes], 114, North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[28]

C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal., 262 (2012), 2777-2832. doi: 10.1016/j.jfa.2012.01.003.  Google Scholar

[29]

C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Manuscripta Math., 146 (2015), 7-63. doi: 10.1007/s00229-014-0684-8.  Google Scholar

[30]

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32, Princeton University Press, Princeton, NJ, 1971.  Google Scholar

[31]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.  Google Scholar

[32]

F. Yao, Global higher integrability for the parabolic equations in Reifenberg domains, Math. Nachr., 283 (2010), 1358-1367. doi: 10.1002/mana.200710084.  Google Scholar

[33]

C. Zhang and S. Zhou, Global weighted estimates for quasilinear elliptic equations with non-standard growth, J. Funct. Anal., 267 (2014), 605-642. doi: 10.1016/j.jfa.2014.03.022.  Google Scholar

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