zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] DOI: 10.1006/jdeq.1999.3665 · Zbl 0945.35010 · doi:10.1006/jdeq.1999.3665
[2] DOI: 10.1007/978-0-8176-4755-1 · doi:10.1007/978-0-8176-4755-1
[3] DOI: 10.1512/iumj.1990.39.39024 · Zbl 0709.35024 · doi:10.1512/iumj.1990.39.39024
[4] —-, Solutions de viscosité des equations de Hamilton-Jacobi (1994)
[5] DOI: 10.1137/0326063 · Zbl 0674.49027 · doi:10.1137/0326063
[6] Bensoussan A., Asymptotic analysis for periodic structures (1978) · Zbl 0404.35001
[7] DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[8] DOI: 10.1090/S0002-9947-1983-0690039-8 · doi:10.1090/S0002-9947-1983-0690039-8
[9] Evans L., Proc. Roy. Soc. Edinburgh Sect. A 120 pp 245– (1992) · Zbl 0796.35011 · doi:10.1017/S0308210500032121
[10] DOI: 10.1512/iumj.1998.47.1385 · Zbl 0924.49020 · doi:10.1512/iumj.1998.47.1385
[11] DOI: 10.1215/S0012-7094-87-05521-9 · Zbl 0697.35030 · doi:10.1215/S0012-7094-87-05521-9
[12] Jikov V. V., Homogenization of differential operators and integral functionals (1994) · doi:10.1007/978-3-642-84659-5
[13] Lions P.-L., Generalized solutions of Hamilton-Jacobi equations (1982)
[14] Lions P.-L., Homogeneization of Hamilton-Jacobi equations (1986)
[15] DOI: 10.1137/0324032 · Zbl 0597.49023 · doi:10.1137/0324032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.