Applications of multipliers to the problem of coercivity in \(W^ l_ p\) of the Neumann problem (Q1207897)

This article deals with the boundary value problem \[ {\partial \over \partial z_ i} \left( a_{ij} (z) {\partial \over \partial z_ j} u \right) = f\quad (z \in \Omega), \quad \text{tr }a_{ij} (z) {\partial \over \partial z_ j} u \cos (\nu,z_ i) = g\quad (z \in \partial \Omega) \] and the Neumann problem for the equation \[ {\partial \over \partial z_ i} \left( a_{ij} (z) {\partial \over \partial z_ j} u \right) + a_ i(z) {\partial \over \partial z_ i} u + a(z)u = f. \] Under natural assumptions about the coefficients in terms of multipliers for Sobolev spaces, coercitive solvability in \(W_ p^ l (\Omega)\) is proved for the first boundary value problem (when \(f \in W_ p^{l - 2} (\Omega)\), \(g \in W_ p^{l - 1 - 1/p} (\partial \Omega)\) and natural or orthogonality conditions for them hold). For the second boundary value problem it is proved that the corresponding differential operator \[ {\mathcal L} : W_ p^{l} (\Omega) \to W_ p^{l - 2} (\Omega) \times W_ p^{l - 1 - 1/p} (\partial \Omega) \] is a Fredholm operator with index zero. Some examples show the essence and exactness of the conditions considered.