Усреднение гармонических 1-форм на псевдоримановых многообразиях сложной микроструктуры

Анотація

4-dimentional manifolds $\widetilde{M}_\varepsilon^4=\mathbf{R}\times M_\varepsilon^3$, where $M_\varepsilon^3$ are Riemannian manifolds of complicated microstructure are considered. $M_\varepsilon^3$ consist of two copies of $\mathbf{R}^3$ with a large number of holes connected in pairs by means of fine tubes. The asymptotic behaviour of harmonic 1-forms on $\widetilde{M}_\varepsilon^4$ is studied as $\varepsilon\to 0$, when the number of tubes on $M_\varepsilon^3$ tends to infinity and their radii tend to zero. The homogenized equations on $\mathbf{R}^4$ describing the leading term of the asymptotics are obtained. The result of homogenization of the solution of Cauchy problem for wave equation on $\widetilde{M}_\varepsilon^4$ as $\varepsilon\to 0$ is obtained.