Asymptotic behavior of nonautonomous monotone and subgradient evolution equations
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- by Hedy Attouch, Alexandre Cabot and Marc-Olivier Czarnecki PDF
- Trans. Amer. Math. Soc. 370 (2018), 755-790 Request permission
Abstract:
In a Hilbert setting $H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $\dot x(t)+A_t(x(t))\ni 0$, where for each $t\geq 0$, $A_t:H\rightrightarrows H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $x(\cdot )$ to a zero of a limit maximal monotone operator $A_\infty$ as the time variable $t$ tends to $+\infty$. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $\mathrm {gph} A_\infty$ over $\mathrm {gph} A_t$ tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators $A_t= \partial \varphi _t$ which are subdifferentials of proper closed convex functions $\varphi _t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.References
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Additional Information
- Hedy Attouch
- Affiliation: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France
- Email: hedy.attouch@univ-montp2.fr
- Alexandre Cabot
- Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France
- MR Author ID: 696546
- Email: alexandre.cabot@u-bourgogne.fr
- Marc-Olivier Czarnecki
- Affiliation: Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier cedex 5, France
- Email: marco@math.univ-montp2.fr
- Received by editor(s): July 30, 2015
- Received by editor(s) in revised form: April 18, 2016
- Published electronically: July 13, 2017
- Additional Notes: The first author’s effort was sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant No. FA9550-14-1-0056
The authors were supported by ECOS grant No. C13E03 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 755-790
- MSC (2010): Primary 34G25, 37N40, 46N10, 47H05
- DOI: https://doi.org/10.1090/tran/6965
- MathSciNet review: 3729487