American Institute of Mathematical Sciences

Advanced
July  2008, 20(3): 617-637. doi: 10.3934/dcds.2008.20.617

$W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains

Sun-Sig Byun 1, and  Lihe Wang 2,

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747
2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received  January 2007 Revised  July 2007 Published  December 2007

We obtain an optimal $W^{1,p}$, $2 \leq p < \infty$, regularity theory on the conormal derivative problem for a nonlinear parabolic equation in divergence form with small BMO nonlinearity in a $\delta$-Reifenberg flat domain.
Keywords: Parabolic maximal function, Conormal derivative problem, Nonlinear Parabolic Equation, BMO space, Reifenberg flat domain..
Mathematics Subject Classification: Primary 35R05, 35R35; Secondary 35J15, 35J2.
Citation: Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617
[1]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4567-4592. doi: 10.3934/dcds.2021049
[2]

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
[3]

Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1629-1645. doi: 10.3934/dcdsb.2021104
[4]

Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024
[5]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
[6]

G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure and Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789
[7]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088
[8]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[9]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[10]

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
[11]

Changchun Liu, Zhao Wang. Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1087-1104. doi: 10.3934/cpaa.2014.13.1087
[12]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[13]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[14]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[15]

Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657
[16]

Vahagn Nersesyan. Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension. Mathematical Control and Related Fields, 2021, 11 (2) : 237-251. doi: 10.3934/mcrf.2020035
[17]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060
[18]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[19]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[20]

Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations and Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (102)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy