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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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