The correct solvability of discontinuous Dirichlet problems in spaces \(L_p\) for elliptic equations (Q2768766)

In the bounded domain \(G\subset\mathbb R^N\) with the boundary consisting of two pieces \(\partial G_1\) and \(\partial G_2\) the author considers the boundary value problem NEWLINE\[NEWLINE\sum_{i,j=1}^{N}\partial_{x_{i}}(a_{ij}(x)\partial x_{j}u)+\sum_{j=1}^{N}b_{j}(x)\partial_{x_{j}}u+c(x)u=f(x),\quad x\in G,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x)=\phi_1(x),\;x\in\partial G_1;\quad u(x)=\phi_2(x),\;x\in\partial G_2.NEWLINE\]NEWLINE The correct solvability of the considered problem is proved in the Liouville space \(L^{\alpha+2}_{p}(G)\), \(p\in(1,\infty)\), in the Besov space \(B_{p,\theta}^{\alpha+2}(G)\), \(\theta\in[1,\infty)\), in the Nikolskii space \(H_{p}^{\alpha+2}(G)\), \(p\in[1,\infty)\), and in Zygmund-Hölder space \(H_{\infty}^{\alpha+2}(G)\).