In 1978 De Giorgi formulated the following conjecture. Let u be a solution of u = u 3 ? u in all of R n such that juj 1 and @ n u > 0 in R n. Is it true that all level sets fu = g of u areâ€¦ (More)

We develop a theory of currents in metric spaces which extends the classical theory of Federerâ€“Fleming in euclidean spaces and in Riemannian manifolds. The main idea, suggested in [20, 21], is toâ€¦ (More)

This paper is concerned with the fine properties of monotone functions on R. We study the continuity and differentiability properties of these functions, the approximability properties, the structureâ€¦ (More)

â–¶ The Journal of Materials Science: Materials in Medicine is the official journal of the European Society for Biomaterials (ESB) â–¶ Covers the science and technology of biomaterials and theirâ€¦ (More)

We consider a variational approach to the problem of recovering missing parts in a panchromatic digital image. Representing the image by a scalar function u, we propose a model based on theâ€¦ (More)

This paper studies a conjecture made by E. De Giorgi in 1978 concerning the onedimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations âˆ†uâ€¦ (More)

In this paper we study the singular perturbation of âˆ« (1âˆ’|âˆ‡u|2)2 by Îµ2|âˆ‡2u|2. This problem, which could be thought as the natural second order version of the classical singular perturbation of theâ€¦ (More)

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimalâ€¦ (More)