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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces


Authors: Moshe Marcus and Victor J. Mizel
Journal: Trans. Amer. Math. Soc. 228 (1977), 1-45
MSC: Primary 46E35; Secondary 46G99
MathSciNet review: 0454622
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Abstract: An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace $ S = {\ring{W}}_k^p[a,b]$ of the kth order Sobolev space $ W_k^p[a,b], 1 \leqslant k, 1 \leqslant p \leqslant \infty $. It is shown that every mapping $ T:S \to {L^1}[a,b]$ which is local, continuous and $ {D^k}$-disjointly additive has the form $ (Tu)(t) = H(t,{D^k}u(t))$, where $ H:[a,b] \times R \to R$ is a function obeying Carathéodory conditions as well as $ (\ast )H( \cdot ,0) = 0$. Here $ {D^k}$-disjoint additivity means $ T(u + v) = Tu + Tv$ whenever $ ({D^k}u)({D^k}v) = 0$. Likewise, every real functional N on S which is continuous and $ {D^k}$-disjointly additive has the form $ N(u) = \smallint Tu$, with T as above. Liapunov's theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0454622-4
PII: S 0002-9947(1977)0454622-4
Keywords: $ {D^k}$-disjointly additive functional, local $ {D^k}$-disjointly additive operator, normalized Carathéodory function, Liapunov vector measure, Radon-Nikodým type theorem
Article copyright: © Copyright 1977 American Mathematical Society