On the $\Gamma$-limit for a non-uniformly bounded sequence of two phase metric functionals
| Autor: | Johannes Zimmer, Hartmut Schwetlick, Daniel C. Sutton |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Large class
Hamiltonian mechanics Pure mathematics Applied Mathematics Mathematical analysis Differentiable manifold 49J45 53C60 symbols.namesake Mathematics - Analysis of PDEs Differential geometry Maupertuis' principle symbols Discrete Mathematics and Combinatorics Uniform boundedness Boundary value problem Hamiltonian (quantum mechanics) Analysis Mathematics |
| Zdroj: | Schwetlick, H, Sutton, D C & Zimmer, J 2015, ' On the Γ-limit for a non-uniformly bounded sequence of two-phase metric functionals ', Discrete and Continuous Dynamical Systems-Series A, vol. 35, no. 1, pp. 411-426 . https://doi.org/10.3934/dcds.2015.35.411 |
| DOI: | 10.3934/dcds.2015.35.411 |
| Popis: | In this study we consider the $\Gamma$-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either $\{1,\infty\}$ or $\{1,\beta \varepsilon^{-p}\}$ where $\beta,\varepsilon > 0$ and $p \in (0,\infty)$. We find that for a large class of metrics, in particular those metrics whose surface of discontinuity forms a differentiable manifold, the $\Gamma$-limit exists, as in the uniformly bounded case. However, when one attempts to determine the $\Gamma$-limit for the corresponding boundary value problem, the existence of the $\Gamma$-limit depends on the value of $p$. Specifically, we show that the power $p=1$ is critical in that the $\Gamma$-limit exists for $p < 1$, whereas it ceases to exist for $p \geq 1$. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential. Comment: 31 pages, 1 figure. Submitted |
| Databáze: | OpenAIRE |
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