Nontrivial solvability of elliptic equations in divergence form with complex coefficients (Q404713)

The paper deals with the stationary Fokker-Planck-Kolmogorov equation with complex diffusion coefficients and a complex vector-field on the torus \(T=[0,1]^n\) with glued opposite faces. Precisely, the equations \[ \sum_{l,k=1}^n \dfrac{\partial}{\partial x_k}\left( a_{l,k}(\mathrm{\mathbf x})\dfrac{\partial U(\mathrm{\mathbf x})}{\partial x_l}\right)=\mathrm{div\,} \mathrm{\mathbf F}(\mathrm{\mathbf x}) \] and \[ \sum_{l,k=1}^n \dfrac{\partial}{\partial x_k}\left( a_{l,k}(\mathrm{\mathbf x})\dfrac{\partial U(\mathrm{\mathbf x})}{\partial x_l}\right)-\mathrm{div\,} \big(U(\mathrm{\mathbf x})\mathrm{\mathbf f}(\mathrm{\mathbf x})\big)=0 \] are examined, with complex coefficients \(a_{l,k}(\mathrm{\mathbf x})\) and with vector fields \(\mathrm{\mathbf F},\;\mathrm{\mathbf f}:\;T\to \mathbb{C}^n.\) Under appropriate conditions on the diffusion coefficients, the author proves nontrivial solvability and shows that the solution space is multidimensional in some cases.