Volume 3, Issue 2

We prove regularity results for solutions of the non-degenerate p -Laplacian on the Heisenberg group ℍ n where μ > 0 and p < 2.

We consider a brittle material under displacement control with a mode III crack running in quasi-static regime along a straight line. Several definitions of propagation, based on variational criteria, have been proposed in the last decade. Our aim is a detailed study of their properties, in particular as far as energy, energy release rate and jump discontinuities is concerned. In order to make similarities and differences appear more clearly, the analysis of the evolutions follows a common structure and theoretical results are illustrated by a prototype numerical example.

We consider Cheeger-like shape optimization problems of the form Min{|Ω| α J (Ω) : Ω ⊂ D } where D is a given bounded domain and α is above the natural scaling. We show the existence of a solution and analyze as J (Ω) the particular cases of the compliance functional C (Ω) and of the first eigenvalue λ 1 (Ω) of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.