Quasilinear divergence form parabolic equations in Reifenberg flat domains

Previous Article
Estimates of the derivatives of minimizers of a special class of variational integrals
DCDS Home
This Issue
Next Article
Solving strongly monotone variational and quasi-variational inequalities

December 2011, 31(4): 1397-1410. doi: 10.3934/dcds.2011.31.1397

Quasilinear divergence form parabolic equations in Reifenberg flat domains

Dian Palagachev ^1, and Lubomira G. Softova ^2,

1.	Department of Mathematics, Polytechnic University of Bari, 4 E. Orabona Str., 70 125 Bari
2.	Dipartimento di Ingegneria Civile, Seconda Università di Napoli, Via Roma, 29; 81 031 Aversa, Italy

Received February 2010 Revised September 2010 Published September 2011

Abstract
Related Papers

We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy--Dirichlet problem for divergence form quasilinear parabolic equations $$ \begin{equation} \left\{\begin{array}{l} u_t-\mathrm{div\,}\big(a^{ij}(x,t,u)D_ju+a^i(x,t,u)\big)=b(x,t,u,Du) &\quad \text{in}\ Q,\\ u=0 &\quad \text{on}\ \partial_p Q, \end{array} \right. \end{equation} $$ where $Q$ is a cylinder in $\mathbb{R}^n\times(0,T)$ with Reifenberg flat base $\Omega.$ The principal coefficients $a^{ij}(x,t,u)$ of the uniformly parabolic operator are supposed to have small $BMO$ norms with respect to $(x,t)$ while the nonlinear terms $a^i(x,t,u)$ and $b(x,t,u,Du)$ support controlled growth conditions.

Keywords: Weak solvability, $ Controlled growth., Reifenberg flat domain, Divergence form quasilinear parabolic equations, Small $BMO, Discontinuous coefficients.

Mathematics Subject Classification: Primary: 35K59; Secondary: 35K61, 35B65, 35R0.

Citation: 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

References:

[1]	A. A. Arkhipova, $L_p$-estimates of the gradients of solutions of initial/boundary-value problems for quasilinear parabolic systems. Differential and pseudodifferential operators, J. Math. Sci., 73 (1995), 609-617. doi: 10.1007/BF02364939.
[2]	A. A. Arkhipova, Reverse Hölder inequalities with boundary integrals and $L_p$-estimates for solutions of nonlinear elliptic and parabolic boundary-value problems, in "Nonlinear Evolution Equations" (ed. N. N. Uraltseva), Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, (1995), 15-42.
[3]	S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301. doi: 10.1007/s00205-005-0357-6.
[4]	S.-S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains, J. Funct. Anal., 223 (2005), 44-85. doi: 10.1016/j.jfa.2004.10.014.
[5]	S.-S. Byun, Optimal $W^{1,p}$ regularity theory for parabolic equations in divergence form, J. Evol. Equ., 7 (2007), 415-428. doi: 10.1007/s00028-007-0278-y.
[6]	S.-S. Byun and L. Wang, Parabolic equations in time dependent Reifenberg domains, Adv. Math., 212 (2007), 797-818. doi: 10.1016/j.aim.2006.12.002.
[7]	O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
[8]	G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[9]	A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Research, 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000.
[10]	J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.
[11]	D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 347 (1995), 2481-2493. doi: 10.2307/2154833.
[12]	D. K. Palagachev, L. Recke and L. G. Softova, Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients, Math. Ann., 336 (2006), 617-637. doi: 10.1007/s00208-006-0014-x.
[13]	E. R. Reifenberg, Solution of the Plateau problem for $m$-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92. doi: 10.1007/BF02547186.
[14]	T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.

show all references

References:

[1]	A. A. Arkhipova, $L_p$-estimates of the gradients of solutions of initial/boundary-value problems for quasilinear parabolic systems. Differential and pseudodifferential operators, J. Math. Sci., 73 (1995), 609-617. doi: 10.1007/BF02364939.
[2]	A. A. Arkhipova, Reverse Hölder inequalities with boundary integrals and $L_p$-estimates for solutions of nonlinear elliptic and parabolic boundary-value problems, in "Nonlinear Evolution Equations" (ed. N. N. Uraltseva), Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, (1995), 15-42.
[3]	S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal., 176 (2005), 271-301. doi: 10.1007/s00205-005-0357-6.
[4]	S.-S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains, J. Funct. Anal., 223 (2005), 44-85. doi: 10.1016/j.jfa.2004.10.014.
[5]	S.-S. Byun, Optimal $W^{1,p}$ regularity theory for parabolic equations in divergence form, J. Evol. Equ., 7 (2007), 415-428. doi: 10.1007/s00028-007-0278-y.
[6]	S.-S. Byun and L. Wang, Parabolic equations in time dependent Reifenberg domains, Adv. Math., 212 (2007), 797-818. doi: 10.1016/j.aim.2006.12.002.
[7]	O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
[8]	G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[9]	A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Research, 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000.
[10]	J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.
[11]	D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 347 (1995), 2481-2493. doi: 10.2307/2154833.
[12]	D. K. Palagachev, L. Recke and L. G. Softova, Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients, Math. Ann., 336 (2006), 617-637. doi: 10.1007/s00208-006-0014-x.
[13]	E. R. Reifenberg, Solution of the Plateau problem for $m$-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92. doi: 10.1007/BF02547186.
[14]	T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.

[1]	Gary Lieberman. Nonlocal problems for quasilinear parabolic equations in divergence form. Conference Publications, 2003, 2003 (Special) : 563-570. doi: 10.3934/proc.2003.2003.563
[2]	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
[3]	Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[4]	M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819
[5]	Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262
[6]	Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
[7]	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
[8]	Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[9]	Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349
[10]	José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002
[11]	Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254
[12]	Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure and Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
[13]	Takahiro Hashimoto. Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients. Conference Publications, 2011, 2011 (Special) : 643-652. doi: 10.3934/proc.2011.2011.643
[14]	M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129
[15]	Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213
[16]	Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure and Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241
[17]	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
[18]	Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613
[19]	Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1413-1429. doi: 10.3934/cpaa.2021026
[20]	Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Dian Palagachev Lubomira G. Softova
on Google Scholar
Dian Palagachev Lubomira G. Softova

[Back to Top]