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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework, https://www.mittagleffler.se/preprints/list.php?program_code=1314f.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework, https://www.mittagleffler.se/preprints/list.php?program_code=1314f.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[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.

[1]

Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038

[2]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[3]

Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155

[4]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems and Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

[5]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026

[6]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991

[7]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939

[8]

Frank Hettlich. The domain derivative for semilinear elliptic inverse obstacle problems. Inverse Problems and Imaging, 2022, 16 (4) : 691-702. doi: 10.3934/ipi.2021071

[9]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure and Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233

[10]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems and Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008

[11]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49

[12]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021

[13]

Henrik Garde, Nuutti Hyvönen. Reconstruction of singular and degenerate inclusions in Calderón's problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022021

[14]

Kim Knudsen, Aksel Kaastrup Rasmussen. Direct regularized reconstruction for the three-dimensional Calderón problem. Inverse Problems and Imaging, 2022, 16 (4) : 871-894. doi: 10.3934/ipi.2022002

[15]

Dian Palagachev, Lubomira G. Softova. Quasilinear divergence form parabolic equations in Reifenberg flat domains. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1397-1410. doi: 10.3934/dcds.2011.31.1397

[16]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019

[17]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics and Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005

[18]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046

[19]

María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049

[20]

N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (164)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]