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Integral representation of convex functions on a space of measures. (English) Zbl 0662.46009

“In duality pairs such as (\({\mathcal M}^ b,{\mathcal C}_ 0)\) and (W\({}^{- 1,p'},W_ 0^{1,p})\) a convex integral functional on the space of functions has a polar which admits an integral representation. This representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part. The duality is useful in plasticity theory. In the Sobolev case the study of non- parametric integrands is new. A description of the sub-differential is obtained.” (Authors’ summary)
The functional in question is the lower semicontinuous hull of the function \[ I_ f(v)=\int f(x,v(x))dx \] on \(L^ 1\) where f(x,\(\cdot)\) is convex with linear growth. Many particular cases are discussed.
Reviewer: H.v.Weizsäcker

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
49L99 Hamilton-Jacobi theories
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
46G05 Derivatives of functions in infinite-dimensional spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
58E30 Variational principles in infinite-dimensional spaces
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