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- Luigi Ambrosio
- 2004

PROPRIETA' DI SEMIGRUPPO In these notes I would like to describe informally the main results obtained in [6], together with some recent improvements. In that paper I study the well posedness of the continuity equation and of the ODE for vector elds b(t; x) having a low regularity with respect to the spatial variables, precisely a BV (bounded variation)… (More)

We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [6, 13] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d −… (More)

We develop a theory of currents in metric spaces which extends the classical theory of Federer–Fleming in euclidean spaces and in Riemannian manifolds. The main idea, suggested in [20, 21], is to replace the duality with differential forms with the duality with (k+ 1)-ples (f, π1, . . . , πk) of Lipschitz functions, where k is the dimension of the current.… (More)

- Luigi Ambrosio
- 2000

This paper is concerned with the study of bounded solutions of semilinear elliptic equations ∆u − F ′(u) = 0 in the whole space R, under the assumption that u is monotone in one direction, say, ∂nu > 0 in R. The goal is to establish the one-dimensional character or symmetry of u, namely, that u only depends on one variable or, equivalently, that the level… (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 structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a C convex… (More)

can be stated and have a solution. Here 0 c R ' is a bounded open set, a, # > 0 and ),-i is the Hausdorff (n 1)-dimensional outer measure in Rn. For instance, when n = 2 and f represents the 2-dimensional image given by a camera, problem (1) can be seen as the weak formulation of the problem of finding the 'best" decomposition of 0 in a finite number of… (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 ∆u = F (u) in all of Rn. We extend to all nonlinearities F ∈ C the symmetry result in dimension n = 3 previously established by the second and the third authors… (More)

Existence theory for a new class of variational problems.

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 the relaxation of the energy

In this paper we study the singular perturbation of ∫ (1−|∇u|2)2 by ε2|∇2u|2. This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy ∫ (1 − u2)2 by ε2|∇u|2, leads, as in the first order case, to energy concentration effects on hypersurfaces. In the two dimensional case we study… (More)