Relaxation for Partially Coercive Integral Functionals with Linear Growth

A Relaxation Result for Autonomous Integral Functionals with Discontinuous Non-coercive Integrand

a L∗∗(y(t), y′(t)) dt : y ∈ AC ([a, b],RN) , y(a) = A, y(b) = B } · (P ∗∗) It is well known that inf F = inf F ∗∗ if L is super-linear and continuous. Recently Cellina in [5] proved that the same conclusion holds true assuming, instead of superlinearity, a weaker growth condition that we will call (GA). Roughly, a convex function L(x, ξ) satisfies (GA) if the intersection of the supporting hype...

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure Spaces

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined through relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that i...

Minima of Non Coercive Functionals

In this paper we study a class of integral functionals defined on H 0 (Ω), but non coercive on the same space, so that the standard approach of the Calculus of Variations does not work. However, the functionals are coercive on W 1,1 0 (Ω) and we will prove the existence of minima, despite the non reflexivity of W 1,1 0 (Ω), which implies that, in general, the Direct Methods fail due to lack of ...

Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings

In this paper we study the lower semicontinuous envelope with respect to the L-topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into RN that are constrained to take values into a smooth submanifold Y of RN . Let B be the unit ball in R and Y a smooth Riemannian manifold of dimension M ≥ 1, isometrically embedded in R for some N ≥ 2....

The Cauchy problem for linear growth functionals