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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

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, 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.  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 "Nonlinear Evolution Equations" (ed. N. N. Uraltseva), Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, (1995), 15-42.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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., Providence, R.I., 1967.  Google Scholar

[8]

G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 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, Berlin, 2000.  Google Scholar

[10]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[11]

D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 347 (1995), 2481-2493. 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), 617-637. 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), 1-92. doi: 10.1007/BF02547186.  Google Scholar

[14]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.  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), 609-617. 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 "Nonlinear Evolution Equations" (ed. N. N. Uraltseva), Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, (1995), 15-42.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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., Providence, R.I., 1967.  Google Scholar

[8]

G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 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, Berlin, 2000.  Google Scholar

[10]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar

[11]

D. K. Palagachev, Quasilinear elliptic equations with $VMO$ coefficients, Trans. Amer. Math. Soc., 347 (1995), 2481-2493. 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), 617-637. 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), 1-92. doi: 10.1007/BF02547186.  Google Scholar

[14]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094.  Google Scholar

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