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$L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary $\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q): u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem $$ \int_Q A(x)\nabla u ...