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Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions. (English) Zbl 1139.35311
Summary: This paper concerns with two issues.
Let $$\Omega\subset\mathbb R^ n$$ be a bounded domain or a half space and $$F(x,\nabla u,\nabla^ 2u)$$ be the fully nonlinear, uniformly elliptic Hamilton-Jacobi-Bellman operator $F(x,\nabla u,\nabla^ 2u):=\sup_ {\alpha\in A}\left\{-\sum^ n_ {i,j=1}a_ {ij}{}^ \alpha(x)\partial^ 2u/(\partial x_ i\partial x_ j)-\sum^ n_ {i=1}b_ i{}^ \alpha(x)\partial u/\partial x_ i\right\},$ where $$A$$ is a set of controls.
The first issue is the existence and the uniqueness of the ergodic type number $$d$$ which appears in the oblique boundary condition $\begin{gathered} F(x,\nabla u,\nabla^ 2u)=0\quad\text{in }\Omega,\\ d+\langle\nabla u,\gamma(x)\rangle-g(x)=0\quad\text{ on }\partial\Omega.\end{gathered}$ .
The second issue is the application of the number for the study of homogenizations of oscillating Neumann boundary conditions $F(x,\nabla u_ \epsilon,\nabla^ 2 u_ \epsilon)=0$ in the domain $\Omega_ \epsilon=\{(x_ 1,x_ 2):\;-a\leq x_ 1\leq a,f_ 0(x_ 1)+\epsilon f_ 1(x_ 1,x_ 1/\epsilon)\leq x_ 2\leq b\subset\mathbb R^ 2\},$ $\langle\nabla u_ \epsilon,\mathbf n_ \epsilon\rangle+c(x_ 1,x_ 1/\epsilon) u_ \epsilon=g(x_ 1,x_ 1/\epsilon)$ on $\Gamma_ \epsilon=\{(x_ 1,x_ 2)\colon\;-a\leq x_ 1\leq a,x_ 2=f_ 0(x_ 1)+\epsilon f_ 1(x_ 1,x_ 1/\epsilon)\},$ $u_ \epsilon=0\quad\text{on }\partial\Omega_ \epsilon\setminus\Gamma_ \epsilon.$ The limit of $$u_ \epsilon$$ as $$\epsilon$$ goes to zero is identified.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35G30 Boundary value problems for nonlinear higher-order PDEs
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