pde and monge-kantorovich mass transfer survey(evans)

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1、PartialDifferentialEquationsandMonge–KantorovichMassTransferLawrenceC.Evans∗DepartmentofMathematicsUniversityofCalifornia,BerkeleySeptember,2001version1.Introduction1.1Optimalmasstransfer1.2Relaxation,dualityPartI:Cost=1(Distance)222.Heuristics2.1Geometryofoptimaltransport2.2Lagrangemulti

2、pliers3.Optimalmasstransport,polarfactorization3.1Solutionofdualproblem3.2Existenceofoptimalmasstransferplan3.3Polarfactorizationofvectorfields4.Regularity4.1SolvingtheMonge–Ampereequation4.2Examples4.3Interiorregularityforconvextargets4.4Boundaryregularityforconvexdomainandtarget5.Applic

3、ation:Nonlinearinterpolation6.Application:Time-stepminimizationandnonlineardiffusion6.1Discretetimeapproximation6.2Euler–Lagrangeequation∗SupportedinpartbyNSFGrantDMS-94-24342.ThispaperappearedinCurrentDevelopmentsinMathematics1997,ed.byS.T.Yau16.3Convergence7.Application:Semigeostrophicm

4、odelsinmeteorology7.1ThePDEinphysicalvariables7.2ThePDEindualvariables7.3FrontogenesisPartII:Cost=Distance8.Heuristics8.1Geometryofoptimaltransport8.2Lagrangemultipliers9.Optimalmasstransport9.1Solutionofdualproblem9.2Existenceofoptimalmasstransferplan9.3Detailedmassbalance,transportdens

5、ity10.Application:Shapeoptimization11.Application:Sandpilemodels11.1Growingsandpiles11.2Collapsingsandpiles11.3Astochasticmodel12.Application:CompressionmoldingPartIII:Appendix13.Finite-dimensionallinearprogrammingReferencesInMemoryofFrederickJ.Almgren,Jr.andEugeneFabes21IntroductionThes

6、enotesareasurveydocumentinganinterestingrecenttrendwithinthecalculusofvariations,theriseofdifferentialequationstechniquesforMonge–Kantorovichtypeoptimalmasstransferproblems.Iwilldiscussinsomedetailanumberofrecentpapersonvariousaspectsofthisgeneralsubject,describingnewlyfoundapplicationsin

7、thecalculusofvari-ationsitselfandinphysics.Animportantthemewillbetheratherdifferentanalyticandgeometrictoolsfor,andphysicalinterpretationsof,Monge–Kantorovichproblemswithauniformlyconvexcostdensity(hereexemplifiedbyc(x,y)=1

8、x−y

9、2)versusthoseprob-2lemswithanonuniformlyconvex