ISSN:
1573-2878
Keywords:
Calculus of variations
;
nonconvex problems
;
Baire category
;
Liapunov theorem
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This paper is concerned with the existence of a minimum in a Sobolev space for the functional $$F_g (x): = \int_0^T {g(x(t)){\text{ }}dt + \int_0^T {h(x'(t)){\text{ }}dt,{\text{ }}x(0) = a,{\text{ }}x(T) = b,} }$$ wherea,b are real numbers,g is a continuous map, andh is lower semicontinuous, satisfying adequate growth conditions. As shown by Cellina and Mariconda, there exists a dense subset of the space of continuous functions bounded below such that, forg in this subset, the above functional attains its minimum no matter whichh is used. This subset contains in particular the monotone maps, as shown by Marcellini, and the concave maps, as shown by Cellina and Colombo. Our aim is twofold: first to show that this subset, although dense, is meager in the sense of the Baire category; and second to show that it contains a class of functions which we call concave-monotone, because it generalizes both the classes of concave and of monotone functions.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02192088
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