| Summary: | 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 $\R^N$:
$$
\begin{cases}
-\dive(A(x)\nabla u) + \lambda = H(x, \nabla u) \qquad \hbox{in $\Omega$,} &
\\
A(x) \nabla u\cdot \vec n=0\qquad \hbox{on $\partial \Omega$,} &
\end{cases}
$$
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 $b(x)\in \elle N$ and $f(x) \in \elle m$, $m\geq 1$).
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 $\elle m$ (or in the dual space $(\acca)'$), previous results which were proved in the literature for bounded solutions and possibly classical or viscosity formulations.
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