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

We consider the so-called 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 R-N:{-div(A(x)del u) + lambda = H(x, del u) in Omega, A(x) del u . (n) over right arrow = 0 on partial derivative Omega,where A(x) is a coercive matrix with bounded coefficients, and H(x, del u) has Lipschitz growth in the gradient and measurable x-dependence with suitable growth in some Lebesgue space (typically, vertical bar H(x, del u)vertical bar <= b(x)vertical bar del u vertical bar + f (x) for functions b(x) is an element of L-N(Omega) and f (x) is an element of L-m(Omega), m >= 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-m(Omega) (or in the dual space (H-1(Omega))'), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.