Erschienen in:

1985 | OriginalPaper | Buchkapitel

Duality, Conjugate Functionals, Monotone Operators and Elliptic Differential Equations

verfasst von : Eberhard Zeidler

Erschienen in: Nonlinear Functional Analysis and its Applications

Verlag: Springer New York

Enthalten in: Professional Book Archive

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In Chapter 49 we showed how one arrives at general duality propositions knowing a Lagrange function L. In this chapter, given a functional F, we define a so-called conjugate functional F*, and in Section 51.4 we explain how one can construct a Lagrange function for a given convex minimum problem with respect to F by means of F*. In this connection, the generalized Young inequality 1a<m:math display='block'> <m:mrow> <m:mi>F</m:mi><m:mo>*</m:mo><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo></m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>≥</m:mo><m:mrow><m:mo>〈</m:mo> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo><m:mo>,</m:mo><m:mi>u</m:mi></m:mrow> <m:mo>〉</m:mo></m:mrow><m:mo>,</m:mo></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$F*\left( {u*} \right) + F\left( u \right) \geqslant \left\langle {u*,u} \right\rangle ,$1b<m:math display='block'> <m:mrow> <m:mi>F</m:mi><m:mo>*</m:mo><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo></m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mrow><m:mo>〈</m:mo> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo><m:mo>,</m:mo><m:mi>u</m:mi></m:mrow> <m:mo>〉</m:mo></m:mrow><m:mo>⇔</m:mo><m:mi>u</m:mi><m:mo>*</m:mo><m:mo>∈</m:mo><m:mo>∂</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$F*\left( {u*} \right) + F\left( u \right) = \left\langle {u*,u} \right\rangle \Leftrightarrow u* \in \partial F\left( u \right)$ and the relation 2<m:math display='block'> <m:mrow> <m:mi>F</m:mi><m:mo>*</m:mo><m:mo>*</m:mo><m:mo>=</m:mo><m:mi>F</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$F** = F$ play a crucial role. One can use 3<m:math display='block'> <m:mrow> <m:msup> <m:mrow> <m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>F</m:mi><m:mo>*</m:mo></m:mrow> <m:mo>)</m:mo></m:mrow></m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo><m:msup> <m:mrow> <m:mrow><m:mo>(</m:mo> <m:msup> <m:mi>F</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>)</m:mo></m:mrow></m:mrow> <m:mrow> <m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> </m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[${\left( {F*} \right)^\prime } = {\left( {F'} \right)^{ - 1}}$ or the more general relation 3a<m:math display='block'> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo><m:mo>∈</m:mo><m:mo>∂</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>⇔</m:mo><m:mi>u</m:mi><m:mo>∈</m:mo><m:mo>∂</m:mo><m:mi>F</m:mi><m:mo>*</m:mo><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi><m:mo>*</m:mo></m:mrow> <m:mo>)</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$u* \in \partial F\left( u \right) \Leftrightarrow u \in \partial F*\left( {u*} \right)$ to calculate F*. Together with the propositions furnished in Chapter 47 concerning subgradients, the calculus of conjugate functionals, whose principal parts are comprised in (la)–(3a), form the “crossing frog” of convex analysis. [For readers who are not familiar with railways, a “crossing frog” is a device on railroad tracks for keeping cars on the proper rails at intersections or switches.]

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