Example of a displacement convex functional of first order



We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.

Mathematics Subject Classification (2000)

35K55 35K30 35K25 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institució Catalana de Recerca i Estudis Avançats (ICREA)BellaterraSpain
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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