Introduction:
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Geometric variational problems arise in various applications e.g. in physics and engineering. Classical examples are the shape of solid bodies minimizing air resistance, area minimizing surfaces with prescribed boundary properties or the shape of membranes giving rise to the lowest fundamental mode. These and other geometric variational problems can be formulated as optimization problems for domain dependent functionals on given classes of subdomains of Riemannian manifolds. Isochoric (volume preserving) or isoperimetric (perimeter preserving) constraints play a fundamental role, and they used to define the relevant classes of domains among which the optimization is performed. Stability estimates (estimating the deviation from the optimal set) have shown several applications, for instance, in the study of variational problems in Riemannian manifolds. In this session, we will also consider related PDEs arising from the Euler-Lagrange equations from the above shape optimization problems (e.g. overdetermined problems, prescibed mean curvature problems) as well as those which share similar mathematical features (e.g. singularly perturbed problems, Allen-Cahn equations). |
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