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Hamilton-Jacobi equations with partial gradient and application to homogenization. (English) Zbl 1014.49021
The authors consider the Hamilton-Jacobi equation $$H(x,u,D_{x'}u)=0$$ in an open set $$\Omega\subset \mathbb{R}^{n}$$ and $$u=g$$ on $$\partial \Omega$$, where the standing variable splits $$x=(x',x'')$$ for $$x'\in\mathbb{R}^{n'}$$ and $$x''\in\mathbb{R}^{n''}$$, $$n'+n''=n$$ and $$D_{x'}u$$ is the partial gradient along $$\mathbb{R}^{n'}\times \{0\}$$ and obtain a uniqueness result for viscosity solutions. Namely, although the equation may have several solutions, they all coincide in an open set of full measure. For special open sets $$\Omega$$ (for instance when $$\Omega =\mathbb{R}^{n}$$), uniqueness holds everywhere. We mention that no compatibility conditions on $$g$$ are assumed. The results are then applied to the homogenization of the Hamilton-Jacobi equation in a perforated set. They yield the a.e. convergence of the solutions of the problem at scale $$\varepsilon$$ as $$\varepsilon \to 0$$ to the solution of the homogenized Hamilton-Jacobi equation.

##### MSC:
 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 35B25 Singular perturbations in context of PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 49L20 Dynamic programming in optimal control and differential games
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