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Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

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  • In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.
    Mathematics Subject Classification: Primary: 35R05, 35J92; Secondary: 35J15, 35J25.


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