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

1 Laboratoire Jacques-Louis Lions, Sorbonne Université, Boîte courrier 187, 4 place Jussieu, 75252 Paris cedex 05, France
2 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

This contribution is part of the Special Issue: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles

Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta

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 $\mathbb{R}^N$ \begin{cases} -div(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)∈ LN(Ω) and f (x) ∈ Lm(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 LN(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.
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