Metric selfduality and monotone vector fields on manifolds

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• 作者：Nassif Ghoussoub^a ; ¹ ; ^{nassif@math.ubc.ca" class="auth_mail" title="E-mail the corresponding author} ; Abbas Moameni^b ; ¹ ; ^{momeni@math.carleton.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Optimal mass transport ; Monotone maps on manifolds ; Variational principles
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：271
• 期：6
• 页码：1652-1690
• 全文大小：554 K

文摘

We develop a “metrically selfdual” variational calculus for c-monotone vector fields between general manifolds X and Y, where c   is a coupling on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000641&_mathId=si1.gif&_user=111111111&_pii=S0022123616000641&_rdoc=1&_issn=00221236&md5=3c9cb2c3f4dba51e5920bb40f0652990" title="Click to view the MathML source">X×Yclass="mathContainer hidden">class="mathCode">. Remarkably, many of the key properties of classical monotone operators known to hold in a linear context extend to this non-linear setting. This includes an integral representation of c-monotone vector fields in terms of c-convex selfdual Lagrangians, their characterization as a partial c-gradients of antisymmetric Hamiltonians, as well as the property that these vector fields are generically single-valued. We also use a symmetric Monge–Kantorovich transport to associate to any measurable map its closest possible c-monotone “rearrangement”. We also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting c-monotone maps, an approach that has proved efficient for resolving non-linear equations and evolutions driven by monotone vector fields in a Hilbertian setting.