Vol. 65, No. 1, 1976

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ISSN: 0030-8730
A characterization of non-linear functionals on W1p possessing autonomous kernels. I

Moshe Marcus and Victor Julius Mizel

Vol. 65 (1976), No. 1, 135–158
Abstract

Let Ω be a domain in Rn and let a nonlinear functional N be given on the first order Sobolev space W1p(Ω), 1 p . We are concerned with obtaining a characterization of those functionals N of the form

       ∫
N (u) =  g(u,D u,⋅⋅⋅ ,D u)dm,  u ∈ W p(Ω ),
1       n             1
(1.1)

where g : Rn+1 R is a continuous function, Diu(i = 1,,n) denotes the distribution derivative of u relative to its i-th coordinate variable and m denotes Lebesgue measure. In the present paper we confine ourselves to the case n = 1. The general case will be considered in the second part of this work.

Mathematical Subject Classification 2000
Primary: 46E35
Milestones
Received: 17 October 1975
Published: 1 July 1976
Authors
Moshe Marcus
Victor Julius Mizel