solvability of linear elliptic pde on a torus

Question

Consider a linear non-divergent form elliptic PDE on a flat torus $\mathbf{T}^n$, $$a_{ij}\partial_{ij}u+b_i\partial_iu=f$$ where all the coefficients and $f$ are smooth. What is the condition that ensures this equation is solvable in $C^{\infty}(\mathbf{T}^n)$, i.e the solutions which are periodic? \ Notice there has to be some condition, as one sees in the case of LaplaciaI, in which the right hand side must have integral zero.

Answer 1

A version of the Fredholm alternative should apply here. Basically, elements of the kernel of the adjoint operator induce conditions on $f$ for your equation to be solvable. Let $(v^k)$ be a basis for the solutions of $\partial_{ij}(a_{ij} v) - \partial_i (b_i v) = 0$. The conditions on $f$ are that $\int_{\mathbf{T}^n} v^k f \, dx = 0$. The Laplacian on the flat torus is self-adjoint and has only one 0-mode, the constant function, which gives you well-known condition on $f$ that you mentioned.

The necessity of these conditions is easy to see: \begin{align} \int_{\mathbf{T}^n} v^k f\, dx &= \int_{\mathbf{T}^n} v^k (a_{ij}\partial_{ij} u + b_i \partial_i u) \, dx \\ &= \int_{\mathbf{T}^n} [\partial_{ij}(a_{ij} v^k) - \partial_i (b_i v^k)] u \, dx = 0 . \end{align}