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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315

[16]

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

[17]

J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379

[18]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[19]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129

[20]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]