The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition

We consider the so-called it ergodic problem for weak solutions of elliptic equations in divergence form, complemented with Neumann boundary conditions. The simplest example reads as the following boundary value problem in a bounded domain of $\mathbb{R}^N$: $ \left\{ \begin{array}{l} - {\rm{div}}(A(x)\nabla u) + \lambda = H(x,\nabla u)\;\;\;\;\;{\rm{in}}\;\Omega ,\\ A(x)\nabla u \cdot \vec n = 0\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega , \end{array} \right. $ where A(x) is a coercive matrix with bounded coefficients, and $H(x, \nabla u)$ has Lipschitz growth in the gradient and measurable $x$-dependence with suitable growth in some Lebesgue space (typically, $|H(x, \nabla u)|\leq b(x) |\nabla u|+ f(x)$ for functions b(x)∈ L^N(Ω) and f (x) ∈ L^m(Ω), $m\geq 1$). We prove that there exists a unique real value $\lambda$ for which the problem is solvable in Sobolev spaces and the solution is unique up to addition of a constant. We also characterize the ergodic limit, say the singular limit obtained by adding a vanishing zero order term in the equation. Our results extend to weak solutions and to data in Lebesgue spaces L^N(Ω) (or in the dual space (H¹(Ω))'), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.