Creating and controlling band gaps in periodic media with small resonators
DOI:
https://doi.org/10.15407/mag19.02.456Анотація
Дослiджуються спектральнi властивостi лапласiана Неймана ${\mathcal A}_\varepsilon$ на перiодичнiй необмеженiй областi ${\Omega}_\varepsilon$ , яка залежить вiд малого параметра $\varepsilon>0$. Область ${\Omega}_\varepsilon$ отримується шляхом видалення з ${\mathbb R}^n$ $m\in{\mathbb N}$ сiмейств $\varepsilon$-перiодично розташованих малих резонаторiв. Доведено, що спектр ${\mathcal A}_\varepsilon$ має принаймнi $m$ лакун. Першi $m$ лакун прямують при $\varepsilon\to 0$ до деяких iнтервалiв, розташуванням i довжиною яких можна керувати шляхом певного вибору резонаторiв; iншi лакуни (якщо вони є) прямують до нескiнченностi. Обговорюються застосування до теорiї фотонних кристалiв.
Mathematical Subject Classification 2020: 35B27, 35P05, 47A75
Ключові слова:
перiодичнi середовища, резонатори, лапласiан Неймана, спектральнi лакуниПосилання
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