Maximal monotone operators are selfdual vector fields and vice-versa

arXivmath/0610494MaRDI QIDQ6478040

Nassif Ghoussoub

Publication date: 16 October 2006

Abstract: If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $Ximes{X}^{*}$, then the vector field $$ is maximal monotone. Conversely, any maximal monotone operator $T$ on $X$ is derived from such a potential on phase space, that is there exists a selfdual Lagrangian $L$ on $Ximes{X}^{*}$ (i.e, $L^{*}(p,x)=L(x,p)$) such that $$. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form $T=partialphi$ for some convex lower semi-continuous function on $X$. This representation allows for the application of the selfdual variational theory --recently developed by the author-- to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form $LambdaxinTx$ for a given map $Lambda:D(Lambda)subsetXo{X}^{*}$, can now be obtained by minimizing functionals of the form $I(x)=L(x,Lambdax)-<x,Lambdax>$.

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