2001/05/16 by Gianni Dal Maso, Maso, Gianni Dal
Mathematics · #35A35 #35B30 #35J25 #35R35 #49Q10 #74R10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A35 #msc:35B30 #msc:35J25 #msc:35R35 #msc:49Q10 #msc:74R10
paper · pdf · doi:10.48550/arxiv.math/0105132
Lecture notes of a course held in the 2001 CNA Summer School ``Multiscale Problems in Nonlinear Analysis'', Carnegie Mellon University, Pittsburgh, May 31--June 9, 2001; 15 pages
arxiv created 2001/05/16 · arxiv updated 2009/11/30
The first part of the course is devoted to the study of solutions to the Laplace equation in Ω∖ K, where Ω is a two-dimensional smooth domain and K is a compact one-dimensional subset of Ω. The solutions are required to satisfy a homogeneous Neumann boundary condition on K and a nonhomogeneous Dirichlet condition on (part of) ∂Ω. The main result is the continuous dependence of the solution on K, with respect to the Hausdorff metric, provided that the number of connected components of K remains bounded. Classical examples show that the result is no longer true without this hypothesis. Using this stability result, the second part of the course develops a rigorous mathematical formulation of a variational quasi-static model of the slow growth of brittle fractures, recently introduced by Francfort and Marigo. Starting from a discrete-time formulation, a more satisfactory continuous-time formulation is obtained, with full justification of the convergence arguments.