Maximal monotone operators are selfdual vector fields and vice-versa
| dc.creator | Ghoussoub, Nassif | |
| dc.date | 2006-10-16 | |
| dc.date.accessioned | 2026-07-07T07:29:07Z | |
| dc.date.available | 2026-07-07T07:29:07Z | |
| dc.description | If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $X\times X^*$, then the vector field $x\to \bar\partial L(x):=\{p\in X^*; (p,x)\in \partial L(x,p)\}$ 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 $X\times X^*$ (i.e, $L^*(p, x) =L(x, p)$) such that $T=\bar\partial L$. 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 $T=\partial ϕ$ 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 $Λx\in Tx$ for a given map $Λ: D(Λ)\subset X\to X^*$, can now be obtained by minimizing functionals of the form $I(x)=L(x,Λx)-< x, Λx>$. | |
| dc.description | 8 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/ | |
| dc.identifier | https://arxiv.org/abs/math/0610494 | |
| dc.identifier | http://arxiv.org/abs/math/0610494 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/117984 | |
| dc.subject | Analysis of PDEs | |
| dc.title | Maximal monotone operators are selfdual vector fields and vice-versa | |
| dc.type | text |