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Duality formulas and variational integrals. (English) Zbl 0789.46038

Considering the finite \(\mathbb{R}^ d\) valued Borel regular measure \(\mu\) on the compact set \(X\subset\mathbb{R}^ N\) and the continuous function \((x,p)\mapsto (x,p): X\times\mathbb{R}^ d\rightarrow\mathbb{R}\) being positively homogeneous of degree one and convex with respect to \(p\), a dual representation for the measure \(f(x,\mu)\) on the Borel sets \(A\subset\mathbb{R}^ n\) according to \[ \int\lim_ A f(x,\mu):=\int\lim_ A f(x,\vec\mu(x))d|\mu| (x)=\int\lim_ A f(x,h(x))d{\mathcal L}^ N \] is proved, where \(| \mu|\) denotes the total variation measure of \(\mu\) and \(\vec\mu(x)=\frac{d\mu}{d|\mu|}(x)\) is the Radon- Nikodym derivative of \(\mu\) with respect to \(|\mu|\). The second equality holds e.g. whenever \(| \mu|\) is absolutely continuous with respect to the Lebesgue measure \({\mathcal L}^ N\) with a density \(h\). For bounded \(|\mu|\)- measurable weight functions \(\varphi: X\rightarrow\mathbb{R}_ +\) it could be shown \[ \begin{split}\int\lim_ X f(x,\mu)\varphi:=\\\sup\{\int\lim_ X<\vec\mu(x),v(x)>\varphi(x) d|\mu|(x);\;v\in{\mathcal C}(X,\mathbb{R}^ d), v(x)\in\partial_ p f(x,0)\forall x\in X\},\end{split} \] where \(\partial_ p f(x,0)\) is the convex subdifferential of \(p\mapsto f(x,p)\) at \(p=0\). The author doesn’t need the inner point condition of T. Rockafellar [Pac. J. Math. 39, 439-469 (1971; Zbl 0236.46031)] for the subdifferential and used instead of the reguarity conditions of Rockafellar for the function \(f^*\) direct assumptions to \(f\).
In the nonhomogeneous case under the increasing property \[ f(x,p)\leq C(1+| p|)\forall(x,p)\in X\times\mathbb{R}^ d \] and the equicontinuity \[ \forall x_ 0,\;\varepsilon>0\;\exists\delta>0\;\forall p\in\mathbb{R}^ d: | x-x_ 0|<\delta\rightarrow| f(x,p)- f(x_ 0,p)|<\varepsilon(1+ | p|) \] of \(f\) the dual representation in analogy to the weighted Fenchel conjugation \[ \begin{split} \int\lim_ X f(x,\mu)\varphi:=\sup\{\int\lim_ X<\vec\mu(x), v(x)>\varphi(x) d|\mu|( x)-\\ \int\lim_ X\varphi(x) f^*(x,v(x))d\alpha;\;v\in{\mathcal C}(X,\mathbb{R}^ d),f^*(\cdot,v(\cdot))\in{\mathcal L}^ 1(X,d\alpha)\}\end{split} \] is shown. For \(f\) independent on \(x\) and with stronger assumptions for \(\varphi\) similar results are obtained by F. Demengel and R. Temam [Indiana Univ. Math. J. 33, 673-709 (1984; Zbl 0581.46036)].
In both cases selection theorems are used. In the homogeneous case the proof was started for measurable \(v(\cdot)\) and then continued by using the approximation of a measurable function by a continuous one. The continuity of the set function \(x\mapsto\partial_ p f(x,0)\) is essential. The last case was proved by the help of the Michael selection theorem, some homogenization method and the extension of homogeneous case where \(X\) is a cone. Nontrivial applications to the representation of \(f(x,\nabla u)\) by the graph of \(u\) are given where \(u\) is of bounded variation.

MSC:

46G10 Vector-valued measures and integration
49J27 Existence theories for problems in abstract spaces
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
26B40 Representation and superposition of functions
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