On the definition and properties of certain variational integrals
1961; American Mathematical Society; Volume: 101; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1961-0138018-9
ISSN1088-6850
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoIn this paper we shall be concerned with certain multiple integrals which arise in the calculus of variations, namely those of the form -f[«] = I f(x, », ux)dx.J R Here R denotes an open set in the number space £n and u = u(x) is a realvalued function defined on R. The integrand f(x, u, p) is assumed to be nonnegative and continuous.Moreover, we suppose throughout the paper that / is convex in p so that l[u] is the integral of a regular variational problem.For the purposes of the calculus of variations, and also for aesthetic reasons, it is natural to want the class of admissible functions u to be as large as possible.Now l[u], as it stands, is certainly well-defined for continuously differentiable functions, but once we go beyond this class there is some question as to the meaning of the integral.If measurable partial derivatives can be associated with u, then one can define l[u] simply as the Lebesgue integral of f(x, u, ux).This procedure cannot be used indiscriminately, however, for it assigns the absurd value ff(x, u, 0)dx to any nonconstant function u whose partial derivatives are zero almost everywhere.As an alternate definition of the integral, we have introduced in [13] a certain lower semicontinuous functional which in general agrees with l[u] whenever u is continuously differentiable, but which at the same time is defined for a much larger class of functions.For convenience in discussing these two integrals the former will be denoted simply by 7[m] and the latter by â[u], (a formal definition of these quantities will be given in §1).Both functionals l[u] and ä[u] are of interest in the calculus of variations; it is the purpose of this paper to clarify the relation between them.An important illustration of the present situation may be found in the theory of area of a nonparametric surface.Indeed, let us denote by Q,[u] the Lebesgue area of a surface z = u(x, y) over a region R in the ordinary (x, y) plane, and set Ç 2 2 1/2 ^ lMJ = I (1 + Mx + uv) dxdy.J R The functional &\u\ then stands in essentially the same relation to A [u] as
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