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 - A, 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.  doi: 10.1016/j.na.2013.11.004.  Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators,, Pure and Applied Mathematics, (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.  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.  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.  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.  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.  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.  doi: 10.1515/anona-2013-0024.  Google Scholar

[11]

A. Friedman, Variational Principles and Free-Boundary Problems,, Second edition, (1988).   Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis,, Graduate Texts in Mathematics, (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.  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, (1993).   Google Scholar

[15]

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

[16]

H. Kempka and J. Vybíral, Lorentz spaces with variable exponents,, Math. Nachr., 287 (2014), 938.  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, (1980).  doi: 10.1137/1.9780898719451.  Google Scholar

[18]

G. Lorentz, Some new functional spaces,, Ann. of Math., 51 (1950), 37.  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.  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.  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.   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.   Google Scholar

[24]

A. Nagurney, Network Economics: A Variational Inequality Approach,, Advances in Computational Economics, (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.  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.  doi: 10.1007/BF02547186.  Google Scholar

[27]

J. Rodrigues, Obstacle Problems in Mathematical Physics,, North-Holland Mathematics Studies, (1987).   Google Scholar

[28]

C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data,, J. Funct. Anal., 262 (2012), 2777.  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.  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, (1971).   Google Scholar

[31]

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

[32]

F. Yao, Global higher integrability for the parabolic equations in Reifenberg domains,, Math. Nachr., 283 (2010), 1358.  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.  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.  doi: 10.1016/j.na.2013.11.004.  Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators,, Pure and Applied Mathematics, (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.  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.  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.  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.  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.  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.  doi: 10.1515/anona-2013-0024.  Google Scholar

[11]

A. Friedman, Variational Principles and Free-Boundary Problems,, Second edition, (1988).   Google Scholar

[12]

L. Grafakos, Modern Fourier Analysis,, Graduate Texts in Mathematics, (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.  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, (1993).   Google Scholar

[15]

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

[16]

H. Kempka and J. Vybíral, Lorentz spaces with variable exponents,, Math. Nachr., 287 (2014), 938.  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, (1980).  doi: 10.1137/1.9780898719451.  Google Scholar

[18]

G. Lorentz, Some new functional spaces,, Ann. of Math., 51 (1950), 37.  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.  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.  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.   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.   Google Scholar

[24]

A. Nagurney, Network Economics: A Variational Inequality Approach,, Advances in Computational Economics, (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.  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.  doi: 10.1007/BF02547186.  Google Scholar

[27]

J. Rodrigues, Obstacle Problems in Mathematical Physics,, North-Holland Mathematics Studies, (1987).   Google Scholar

[28]

C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data,, J. Funct. Anal., 262 (2012), 2777.  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.  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, (1971).   Google Scholar

[31]

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

[32]

F. Yao, Global higher integrability for the parabolic equations in Reifenberg domains,, Math. Nachr., 283 (2010), 1358.  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.  doi: 10.1016/j.jfa.2014.03.022.  Google Scholar

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