# American Institute of Mathematical Sciences

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

## Quasilinear divergence form parabolic equations in Reifenberg flat domains

 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

We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy--Dirichlet problem for divergence form quasilinear parabolic equations $$$$\left\{\begin{array}{l} u_t-\mathrm{div\,}\big(a^{ij}(x,t,u)D_ju+a^i(x,t,u)\big)=b(x,t,u,Du) &\quad \text{in}\ Q,\\ u=0 &\quad \text{on}\ \partial_p Q, \end{array} \right.$$$$ 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.
Citation: 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
##### 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. doi: 10.1007/BF02364939. Google Scholar [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, 164 (1995), 15. Google Scholar [3] S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains,, Arch. Ration. Mech. Anal., 176 (2005), 271. doi: 10.1007/s00205-005-0357-6. Google Scholar [4] S.-S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains,, J. Funct. Anal., 223 (2005), 44. doi: 10.1016/j.jfa.2004.10.014. Google Scholar [5] S.-S. Byun, Optimal $W^{1,p}$ regularity theory for parabolic equations in divergence form,, J. Evol. Equ., 7 (2007), 415. doi: 10.1007/s00028-007-0278-y. Google Scholar [6] S.-S. Byun and L. Wang, Parabolic equations in time dependent Reifenberg domains,, Adv. Math., 212 (2007), 797. doi: 10.1016/j.aim.2006.12.002. Google Scholar [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., (1967). Google Scholar [8] G. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). Google Scholar [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, (2000). Google Scholar [10] J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. doi: 10.2307/2372841. Google Scholar [11] D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients,, Trans. Amer. Math. Soc., 347 (1995), 2481. doi: 10.2307/2154833. Google Scholar [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. doi: 10.1007/s00208-006-0014-x. Google Scholar [13] E. R. Reifenberg, Solution of the Plateau problem for $m$-dimensional surfaces of varying topological type,, Acta Math., 104 (1960), 1. doi: 10.1007/BF02547186. Google Scholar [14] T. Toro, Doubling and flatness: Geometry of measures,, Notices Amer. Math. Soc., 44 (1997), 1087. Google Scholar

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. doi: 10.1007/BF02364939. Google Scholar [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, 164 (1995), 15. Google Scholar [3] S.-S. Byun and L. Wang, Parabolic equations in Reifenberg domains,, Arch. Ration. Mech. Anal., 176 (2005), 271. doi: 10.1007/s00205-005-0357-6. Google Scholar [4] S.-S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains,, J. Funct. Anal., 223 (2005), 44. doi: 10.1016/j.jfa.2004.10.014. Google Scholar [5] S.-S. Byun, Optimal $W^{1,p}$ regularity theory for parabolic equations in divergence form,, J. Evol. Equ., 7 (2007), 415. doi: 10.1007/s00028-007-0278-y. Google Scholar [6] S.-S. Byun and L. Wang, Parabolic equations in time dependent Reifenberg domains,, Adv. Math., 212 (2007), 797. doi: 10.1016/j.aim.2006.12.002. Google Scholar [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., (1967). Google Scholar [8] G. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). Google Scholar [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, (2000). Google Scholar [10] J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. doi: 10.2307/2372841. Google Scholar [11] D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients,, Trans. Amer. Math. Soc., 347 (1995), 2481. doi: 10.2307/2154833. Google Scholar [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. doi: 10.1007/s00208-006-0014-x. Google Scholar [13] E. R. Reifenberg, Solution of the Plateau problem for $m$-dimensional surfaces of varying topological type,, Acta Math., 104 (1960), 1. doi: 10.1007/BF02547186. Google Scholar [14] T. Toro, Doubling and flatness: Geometry of measures,, Notices Amer. Math. Soc., 44 (1997), 1087. Google Scholar
 [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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617 [8] 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 [9] José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002 [10] 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 [11] Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043 [12] 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 [13] M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129 [14] Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213 [15] Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241 [16] Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613 [17] Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004 [18] M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885 [19] Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31 [20] Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213

2018 Impact Factor: 1.143