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Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

Authors:

Qi Kang Ran

Institution:

(1) Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, P. R. China

Abstract:

In this paper, we prove that the weak solutions $u \in W^{{1,p}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {1 < p < \infty } \right)}$ of the following equation with vanishing mean oscillation coefficients A(x):

$- {\text{div}}{\left {{\left( {A{\left( x \right)}\nabla u \cdot \nabla u} \right)}^{{\frac{{p - 2}} {2}}} A{\left( x \right)}\nabla u + {\left| {F{\left( x \right)}} \right|}^{{p - 2}} F{\left( x \right)}} \right]} = B{\left( {x,u,\nabla u} \right)},$

belong to $W^{{1,q}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {\forall q \in {\left( {p,\infty } \right)}} \right)}$ , provided $F \in L^{q}_{{{\text{loc}}}} {\left( \Omega \right)}$ and B(x, u, h) satisfies proper growth conditions, where Ω ⊂ R ^N (N ≥ 2) is a bounded open set, A(x) = (A ^ij(x))_N×N is a symmetric matrix function. This work is supported by National Natural Science Foundation of China (10371021)

Keywords:

Nonlinear elliptic equations Local Regularity Calderón– Zygmund decomposition VMO space Local weak L p (Ω ) space

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