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In these lectures I will discuss two kinds of problems from conformal geometry, with the goal of showing an important connection between them in four dimensions. The first problem is a fully nonlinear version of the Yamabe problem, known as the σk-Yamabe problem. This problem is, in general, not variational (or at least there is not a natural variational interpretation), and the underlying equation...
In this lectures we present a series of results concerning a class of diffusion second order PDE’s of heat-type. The results we show have been obtained in collaboration with M.Bramanti, L.Brandolini and F. Uguzzoni (see [9], [10], [24]). The exended version of the main results presented in these notes is contained in [10].
The purpose of these notes is to present some techniques for constructing solutions to a class of singularly perturbed problems with a precise asymptotic behavior when the perturbation parameter ε tends to zero. We first treat the case of concentration at points, and then the case of concentration at manifolds. One of the main motivations for the study of these equations arises from reaction-diffusion...
In these lectures we use an approach introduced by Taubes (cf. [JT]) in the study of selfdual vortices for the abelian-Higgs model, in order to describe vortex configurations for the Chern-Simons (CS in short) theory discussed in [D1]. Notice that the abelian-Higgs model corresponds (in a non-relativistic context) to the bi-dimensional Ginzburg-Landau (GL in short) model (cf. [GL], 1[DGP]), for which...
The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient...
The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient...
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