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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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