Determination of the potential form of operators

J. J. Telega

Mathematica Applicanda, Tome 10 (1982) no. 18, pp. 107-122.

Résumé

The author extends ideas of duality [see, for example, B. Noble and M. J. Sewell, J. Inst. Math. Appl. 9 (1972), 123–193; MR0307012] to a class of nonlinear operators on Banach spaces. Let U, V be Banach spaces and a(u,v) a bilinear form on U×V. Let N be a (nonlinear) operator N:U→V. GN(u)h denotes the Gâteaux derivative of N in the direction of h, computed at the point u∈U. Let us assume that a separates points in U×V (as defined by Marshall Stone). If there is v∈V such that a(h,v)=〈h,Gf(u)〉 for a functional f:U→R then v is called the gradient of f(u). The operator N is called potential if a suitable functional f satisfying this condition exists. The problem of symmetrizing N involves a suitable choice of the bilinear form a. For example, the operator N(u(t))=[(du/dt)2−g(t)] is not potential with respect to the usual L2 product. The author formulates a number of variational principles and discusses specific examples. This is an interesting article, supplementing the ideas of E. Tonti and of R. W. Atherton and G. M. Homsy [Studies in Appl. Math. 54 (1975), no. 1, 31–60; MR0458271].

Détail

DOI : 10.14708/ma.v10i18.1526

Classification : 49H05(47H99 58E30)
Mots-clés : Variational principles of physics,Variational principles