Quasilinear divergence form parabolic equations in Reifenberg flat domains

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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
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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 & Continuous Dynamical Systems - A, 2011, 31 (4) : 1397-1410. doi: 10.3934/dcds.2011.31.1397

References:

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[8]	G. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). Google Scholar
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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

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