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Explore advanced mathematical concepts in this 41-minute lecture focusing on dimension reduction techniques in thin domains. Delve into fundamental strategies for studying the Dirichlet Laplacian on thin two-dimensional strips, examining three distinct models: straight strips, curved strips, and strips with non-uniform width. Discover how geometric properties influence the asymptotic behavior of eigenvalues as strip width approaches zero, and learn about the resulting effective operators including one-dimensional Laplacian and Schrödinger-type operators with geometric potentials. Extend your understanding to strips embedded in three-dimensional space equipped with mixed Dirichlet–Neumann boundary conditions, building upon previous research in purely Dirichlet cases. Compare and contrast the asymptotic behavior of eigenvalues between different boundary condition settings, highlighting both similarities and differences as strips become increasingly thin. Gain insights into cutting-edge research in mathematical analysis and partial differential equations through this specialized presentation delivered at the Centre International de Rencontres Mathématiques during the thematic meeting on wave propagation in guiding structures.
