Example of a first order displacement convex functional
José Antonio Carrillo, Dejan Slepčev · 2009
Abstract. We present a family of first-order functionals which are displacement convex, that is convex along the geodesics in-duced 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 bound-ary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow. 1.