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Convex functions of a measure and applications. (English) Zbl 0581.46036
The authors consider properties of the real valued functional \(F(\mu)=\int_{\Omega}f(\mu)\) on the space \(M_ 1(\Omega)\) of bounded measures \(\mu\) on \(\Omega\), where \(\Omega\) is an open subset of \({\mathbb{R}}^ n\), and f is a nonlinear continuous function with almost linear growth at infinity. The first two sections provide various general properties of f(\(\mu)\) with no convexity assumption of f. In the convex case, a duality formula is given relating the definition of f(\(\mu)\) to the Legendre transform of f. Specifically, if f is convex, \(f: R^{\ell}\to R\) and \(| f(\xi)| \leq k_ 1(1+| \xi |)\) then for every \(\phi \in b_ 0(\Omega)\), the following holds: \(<f(\mu),\phi >=\sup_{v}\{<\mu,v\phi >-\int_{\Omega}f^*(v)\phi dx\}\) where sup is taken over all \(v\in L_ p(\Omega,\mu)\) with \(1\leq p\leq \infty\) and \(f^*\circ v\in L_ 1(\Omega)\). Here, \(f^*\) denotes the Rockafellar conjugate of f defined by \(f^*(u^*)=\sup_{u}\{<u^*,u>-f(u)\}.\) Among the approximation results proved on F(\(\mu)\) with f convex and \(f: R^{\ell}\to R\), the following is of interest: The space \(b_ 0^{\infty}(\Omega)\) of smooth functions with compact support on \(\Omega\), is dense in \((M_ 1(\Omega))^{\ell}\) with a topology that is intermediate between the weak topology and the norm topology. Convergence in the intermediate topology is defined by: \(\mu_ j\to \mu\) weakly and \(\int_{\Omega}f(\mu_ j)\to \int_{\Omega}f(\mu).\)
Other approximation results involve function spaces that are metric but are not normed. These spaces are related to variational problems in solid mechanics. More specifically, the authors consider weak solutions of the evolution equation \(\partial u/\partial t+Au=0\) where A is the minimal surface differential operator. Using the results of first two sections, the authors consider two classes of variational problems with singular solutions. The first result relates to a model for plastic plates studied by Demengel and the second result derives optimality for a variational problem in the deformation theory for perfect (Hencky) plasticity.
Reviewer: M.Sury

46G05 Derivatives of functions in infinite-dimensional spaces
74S30 Other numerical methods in solid mechanics (MSC2010)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
90C25 Convex programming
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
46A55 Convex sets in topological linear spaces; Choquet theory
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