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Aix-Marseille Université
Institut de Mathématiques de Marseille (I2M) - UMR 7373
Site Saint-Charles : 3 place Victor Hugo, Case 19, 13331 Marseille Cedex 3
Site Luminy : Campus de Luminy - Case 907 - 13288 Marseille Cedex 9

Soutenance de thèse

Positivity and qualitative properties of solutions of fourth-order elliptic equations

Giulio Romani
I2M, Aix-Marseille Université

Date(s) : 10/10/2017   iCal
14h00 - 16h00

This thesis concerns the study of fourth-order elliptic boundary value problems and, in particular, qualitative properties of solutions. Such problems arise in various fields, from plate theory to conformal geometry and, compared to their second-order counterparts, they present intrinsic difficulties, mainly due to the lack of the maximum principle.
In the first part of the thesis, we study the positivity of solutions in case of Steklov boundary conditions, which are intermediate between Dirichlet and Navier boundary conditions. They naturally appear in the study of the minimizers of the Kirchhoff-Love functional, which represents the energy of a hinged thin and loaded plate in dependence of a parameter. We establish sufficient conditions on the domain to obtain the positivity of the minimizers of the functional. Then, for such domains, we study a generalized version of the functional. Using variational techniques, we investigate existence and positivity of the ground states, as well as their asymptotic behaviour for the relevant values of the parameter.
In the second part of the thesis we establish uniform a-priori bounds for a class of fourth-order semilinear problems in dimension 4 with exponential nonlinearities. We considered both Dirichlet and Navier boundary conditions and we suppose our nonlinearities positive and subcritical. Our arguments combine uniform estimates near the boundary and a blow-up analysis. Finally, by means of the degree theory, we obtain the existence of a positive solution.

*Membres du jury :

François Hamel, Aix-Marseille Université (Directeur)
Enea Parini, Aix-Marseille Université (Directeur)
Bernhard Ruf, Università degli Studi di Milano (Co-Directeur)
Filippo Gazzola, Politecnico di Milano (Rapporteur)
Frédéric Robert, Université de Lorraine (Rapporteur)
Elvise Berchio, Politecnico di Torino (Examinatrice)
Anna Dall’Acqua, Universität Ulm (Examinatrice)
Louis Dupaigne, Université de Lyon 1 (Examinateur)

Lien : theses.fr

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