2019/01/24 by Gabriele Cora, Cora, Gabriele
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Applied mathematics #Compact space #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Multiplicity (mathematics) #Nonlinear Partial Differential Equations #Nonlinear system #Numerical methods in inverse problems #Ode #Physics #Variational principle #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1901.08337
Ph.D. Thesis, 122 pages
arxiv created 2019/01/24 · openalex publication_date 2019/01/24 · arxiv updated 2019/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this thesis we deal with two different classes of variational problems: 1) the problem of closed curves with prescribed curvature, or H-loop problem; 2) the study of the nodal solutions of the fractional Brezis-Nirenberg problem. In both cases we deal with nonlinear equations (an ODE system for problem 1, and an elliptic equation for problem 2) which admit a variational structure. Nevertheless, in both cases, a lack of compactness occurs and this constitutes a strong obstruction for the application of the direct methods of the calculus of variations or even standard variational methods. Therefore, we are lead to use more refined techniques or also to follow different approaches, like the Lyapunov-Schmidt method and blow-up analysis. This will allow us not only to obtain existence and multiplicity results, but also qualitative properties of the solutions.