VARIATIONAL PROBLEMS WITH RESPECT TO A MEASURE

Giuseppe Buttazzo

Dipartimento di Matematica Università di Pisa Via Buonarroti, 2 56127 PISA (Italy)

   buttazzo@dm.unipi.it,   http://www.dm.unipi.it/pages/buttazzo/

AMS classification: 49J45, 49Q10, 74P05

Keywords: variational integrals, measures, relaxation.

Abstract.

In the study of thin elastic structures, the usual approach consists in fattening the structure to obtain a 3-dimensional domain of a thickness ε, considering on this domain the 3-dimensional constitutive equations, and passing to the limit as ε → 0. This method has been very fruitful to obtain and justify several models of elastic 1dimensional (bars, beams, rods,…) and 2-dimensional (membranes, shells, plates) structures. Some difficulties of the method appear when the structure is irregular or when junctions of different dimensions arise. This becomes crucial in shape optimization theory, where the regularity of the optimal structures is not a priori known.

Another domain where difficulties of this kind arise is the one of optimal control problems where the control variable is a manifold whose structure, along a minimizing sequence, may become more and more complex.

To overcome these difficulties we present an attempt to treat structures as measures and we introduce for every measure μ the notion of tangent bundle associated to μ. This makes possible to study energy functionals of the form

F_{μ}(u) = ∫ f(x, Du) dμ

and their relaxed representative

F_{μ}(u) = ∫ f_{μ}(x, D_{μ}u) dμ

where the tangential gradient D_{μ} and the relaxed density f_{μ} are suitably defined. An interesting application to shape optimization problems is showed.