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{\bf Example of a First Order Displacement Convex Functional}

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Jos\'e Carrillo\\
ICREA (Instituci\'{o} Catalana de Recerca I Estudis Avan\c{c}ats and\\
Departament de Matem\`{a}tiques, Universitat Aut\`{o}noma de Barcelona\\
E-08193 Bellaterra, Spain\\
carrillo@mat.uab.es


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Dejan Slep\v{c}ev\\
Carnegie Mellon University\\
Department of Mathematical Sciences\\
Pittsburgh, PA  15213\\
slepcev@andrew.cmu.edu
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\noindent {\bf Abstract}:  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.


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