# Relaxation of second order geometric integrals and non-local effects

Anza Hafsa, O; Mandallena, J-P (2004). Relaxation of second order geometric integrals and non-local effects. Journal of Nonlinear and Convex Analysis, 5(3):295-306.

## Abstract

We are concerned with the relaxation of second-order geometric integrals, i.e., functionals of the type:
C c ∞ (ℝ N )∋u↦F μ (u):=∫ ℝ N f∇ 2 u (x)dμ(x),
where ∇ 2 u is the Hessian of u, f:MsymN→[0,+∞] is a continuous function, and μ is a finite positive Radon measure on ℝ N . A relaxation problem of this type was studied for the first time by G. Bouchitté and I. Fragala , where they pointed out a new phenomenon: the functional relaxed of F μ has, in general, a 'non-local' representation. Working on a more formal level than in, we develop an alternative method making clear this 'strange phenomenon'.

We are concerned with the relaxation of second-order geometric integrals, i.e., functionals of the type:
C c ∞ (ℝ N )∋u↦F μ (u):=∫ ℝ N f∇ 2 u (x)dμ(x),
where ∇ 2 u is the Hessian of u, f:MsymN→[0,+∞] is a continuous function, and μ is a finite positive Radon measure on ℝ N . A relaxation problem of this type was studied for the first time by G. Bouchitté and I. Fragala , where they pointed out a new phenomenon: the functional relaxed of F μ has, in general, a 'non-local' representation. Working on a more formal level than in, we develop an alternative method making clear this 'strange phenomenon'.

## Citations

6 downloads since deposited on 29 Nov 2010
Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2004 29 Nov 2010 16:26 05 Apr 2016 13:24 Yokohama 1345-4773 http://www.ybook.co.jp/online/jncae/vol5/num3.htm http://www.ams.org/mathscinet-getitem?mr=2111605