BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Filippo Santambrogio\, (Université Claude Bernard - Lyon 1) DTSTART:20200505T150000Z DTEND:20200505T160000Z DTSTAMP:20260404T121817Z UID:LisbonWADE/3 DESCRIPTION:Title: Optimal transport methods for the regularity of 2D functions of least g radient\nby Filippo Santambrogio\, (Université Claude Bernard - Lyon 1) as part of Lisbon webinar in analysis and differential equations\n\n\nA bstract\nThe least gradient problem (minimizing the BV norm with given bou ndary data)\, motivated by both image processing applications and connecti ons with minimal surfaces\, is known to be equivalent\, in the plane\, to the Beckmann minimal-flow problem (an alternative formulation of the $L^1$ Monge-Kantorovich optimal transport problem) with source and target measu res located on the boundary of the domain. Hence\, Sobolev regularity of f unctions of least gradient is equivalent in this setting to $L^p$ bounds o n the solution of the Beckmann problem (i.e. on what is called the transpo rt density) and can be attacked with techniques which are now standard in optimal transport. From the transport point of view\, the novelty of the e stimates that I will present\, coming from a joint paper with S. Dweik\, l ies in the fact they are obtained for transport between measures which are concentrated on the boundary. From the BV point of view\, a new result is the $W^{1\,p}$ regularity of the least gradient function whenever the bou ndary datum is $W^{1\,p}$ as a 1D function: moreover\, the optimal transpo rt framework is strong enough to deal with arbitrary strictly convex norms instead of the Euclidean one with almost no effort.\n LOCATION:https://researchseminars.org/talk/LisbonWADE/3/ END:VEVENT END:VCALENDAR