### Mathematics in Engineering

2021, Issue 4: 1-20. doi: 10.3934/mine.2021031
Research article Special Issues

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

• Received: 17 January 2020 Accepted: 06 March 2020 Published: 18 August 2020
• 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)∈ 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.

Citation: François Murat, Alessio Porretta. The ergodic limit for weak solutions of elliptic equations with Neumann boundary condition[J]. Mathematics in Engineering, 2021, 3(4): 1-20. doi: 10.3934/mine.2021031

### Related Papers:

• 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)∈ 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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