Summer School in Analysis and PDEs
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Courses
Elvise Berchio (Politecnico di Torino)
Title: Partially Hinged Rectangular Plates: Theoretical Aspects And Applications
Abstract
The aim of the course is to present some relevant aspects in the theory of partially hinged rectangular plates together with their possible role in mathematical models of bridges.
The bending energy of a plate is wellknown from the KirchhoffLove theory of elasticity, we will start by deriving from it the biharmonic equation satisfied by the equilibrium position of the plate together with the associated boundary conditions. This is followed by the introduction of a suitable variational framework that allows us to develop a theoretical study of existence, uniqueness, and regularity of solutions of the boundary value problem obtained. In the course a particular attention will be devoted to illustrate some general theoretical aspects of biharmonic equations together with some crucial difficulties arising when dealing with these equations, such as the failure of truncation and maximum principles. The spectral properties of the biharmonic operator under partially hinged boundary conditions will be also presented relating them to the stability properties of the plate and, in turn, of the bridge modelled by it.
The possible role of partially hinged plates in mathematical models for bridges was pointed out in [4] while a thorough study of related theoretical and applicative aspects has been developed in a series of subsequent papers, see e.g. [1,2,3].
References
[1] E. Berchio, A. Ferrero, F. Gazzola, “Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some longstanding questions”, Nonlinear Anal. Real World Appl. 28 (2016), 91–125
[2] E. Berchio, D. Buoso, F. Gazzola, D. Zucco, “A minimaxmax problem for improving the torsional stability of rectangular plates”, J. Optim. Theory Appl. 177 (2018), no. 1, 64–92
[3] E. Berchio, A. Falocchi, “Maximizing the ratio of eigenvalues of nonhomogeneous partially hinged plates”, Journal Spectr. Theory v.11 (2) (2021), 743780
[4] A. Ferrero, F. Gazzola, “A partially hinged rectangular plate as a model for suspension bridges”, Discrete Contin. Dyn. Syst, 35(12) (2015), 5879–5908
Massimiliano Morini (Università di Parma)
Title: "Weak approaches to level set curvature flows"
Abstract
Among geometric evolution problems the motion of a surface according to its mean curvature is the best known and it has been widely studied in the last four decades. Since singularities may appear in finite time, the classical formulation is in general inadequate to define globalin time solutions. Thus, several weak formulations have been proposed in the literature to overcome this issue and to deal with general nonsmooth initial data. One of the possibilities is to represent the initial "surface" as the level set of an auxiliary (initial) function and then to let evolve all the level sets of such a function according to the same geometric law. This procedure transforms the original geometric evolution problem into an initial value problem for a suitable degenerate parabolic PDE, the socalled level set formulation. We will start by reviewing the classical viscosity solution approach developed independently by EvansSpruck and ChenGigaGoto, and then describe its connection with the minimizing movements scheme proposed by AlmgrenTaylorWang. The above methods are quite general and allow one to treat a large class of generalised, possibly anisotropic curvature motions. However, when the underlying anisotropy is of crystalline type, the associated curvature operator becomes nonlocal and highly degenerate and the above classical approaches fail to apply. In the last part of the course a recent new weak formulation based on distance functions and yielding wellposedness of the level set flow also in the crystalline case will be presented. Finally, as time permits, possible applications of the new method to more general nonlocal motions will be outlined.
Gianmaria Verzini (Politecnico di Milano)
Title "Optimization of eigenvalues in reactiondiffusion models"
Abstract
In population dynamics, a pivotal class of models for the dispersal of a species in a heterogeneous environment is based on logistic reactiondiffusion equations. To describe the environment divided into different zones, a weight that changes sign is introduced, so that the positive weight zones correspond to the zones favoring survival. When the equation is associated with homogeneous boundary conditions, the problem presents a threshold for the persistence of the population, encoded by a principal eigenvalue of the associated linearized equation. The problem of minimization of this eigenvalue therefore appears naturally in this context.
During the course, we will consider various models in this context and the derivation of the related optimization problems, illustrating the main open questions and possible lines of research.
For instance, when the minimization is set on a suitable class of weights, the minimum is reached by a piecewise constant (bangbang) control, and thus the problem translates in a shape optimization/free boundary one. The qualitative properties of the optimal set are completely understood only in the onedimensional case while many questions are open in higher dimensions, and can be attacked, for instance, by singular perturbation analysis.
Similarly, other optimization problems related to the presence of other diffusion operators or even to the presence of more populations competing for survival in the same territory are relevant.
Talks
William Borrelli (Università Cattolica di Brescia)
Azahara de la Torre (“Sapienza” Università di Roma)
Alessandra de Luca (Università Ca' Foscari di Venezia)
Alessio Falocchi (Politecnico di Milano)
Valentina Franceschi (Università di Padova)
Vesa Julin (University of Jyväskylä, Finlandia)
Anna Kubin (Politecnico di Torino)
Dario Mazzoleni (Università di Pavia)
Paolo Musolino (Università di Venezia)
Alessandra Pluda (Università di Pisa)
Gieorgios Psaradakis (Università della Campania "Luigi Vanvitelli")
Alberto Roncoroni (Politecnico di Milano)
Delia Schiera (Instituto Superior Tècnico, Lisboa, Portogallo)
Scientific and organizing committee
Giuseppina di Blasio  Università della Campania "Luigi Vanvitelli"
Benedetta Pellacci  Università della Campania "Luigi Vanvitelli"
Giovanni Pisante  Università della Campania "Luigi Vanvitelli"
The attendance of the school will not guarantee the acquisition of university credits.

V: ALERE VAnvitelli 

