Cetraro, June 11-76, 2007 Geometric aspects of concentration phenomena Prof. Andrea Malchiodi, SISSA, Italia Abstract The course is focused on concentration phenomena for some singularly perturbed elliptic equations with motivations from biology or physics. As a model problem, we study a nonlinear elliptic equation with Neumann boundary conditions on a bounded domain. We first discuss the phenomenon of concentration at points of the domain (at the boundary or at the interior). This is done using a perturbative technique developed by Ambrosetti and Badiale to reduce the problem to a finite-dimensional one, exploiting its variational structure, see 1). We then turn to the case of concentration at higher-dimensional subsets, like the whole boundary of the domain, or its minimal submanifolds (for example, geodesics). These results rely on local inversion arguments combined with Fourier analysis and asymptotic expansions, see 2)-5). Prerequisites are basic knowledge of funcional analysis and elliptic PDEs, and some backgound in differential geometry, see e.g. 6). 1) Antonio Ambrosetti - Andrea Malchiodi: Perturbation Methods and Semilinear Elliptic problems on R^n, Birkhauser, Progress in Mathematics, vol.240, (2005) 2) Andrea Malchiodi - Marcelo Montenegro: Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J. 124 (2004), no. 1, 105-143. 3) Andrea Malchiodi: Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, G.A.F.A., 15-6 (2005), 1162-1222. 4) Fethi Mahmoudi - Andrea Malchiodi: Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. in Math., 209-2 (2007), 460-525. 5) Andrea Malchiodi: Some recent results about a class of singularly perturbed elliptic equations, CRM proceedings, to appear (available at http://people.sissa.it/~malchiod/ ). 6) Do Carmo, Manfredo: Riemannian geometry, Mathematics: Theory & Applications. Birkhuser Boston, Inc., Boston, MA, 1992. |