Executive Summary : | The Laplace operator's eigenvalues, which represent frequencies in musical instruments, are affected by the shapes of strings and drums. This leads to a mathematical question about the relationship between the Laplacian and the geometric structure of the domain under consideration. Obstacle placement problems and isoperimetric problems are examples of such problems. Obstacle placement problems involve finding a position of Ω₁ inside Ω₂ where the functional (eigenvalue) under consideration takes its optimum value. These problems are crucial in the design of liquid crystal devices, musical instruments, and optimal accelerator cavities. Isoperimetric problems optimize eigen-values under various geometrical constraints, and they have applications in applied mathematics and engineering. Eigenvalues are also used in PDEs, dynamical systems, classical mechanics, functional analysis, and probability analysis. This research proposal focuses on studying the mixed Steklov-Dirichlet eigenvalue Problem, focusing on finding isoperimetric bounds for Steklov-Dirichlet eigenvalues on domains in simply connected space forms and obstacle placement problems for the first mixed Steklov-Dirichlet eigenvalue on doubly connected domains in an n-dimensional Euclidean space. |