Browsing by Subject "partial differential equations"

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    Optimal transport, free boundary regularity, and stability results for geometric and functional inequalities
    (2013-05) Indrei, Emanuel Gabriel; Figalli, Alessio; Caffarelli, Luis A; Gamba, Irene M; Beckner, William; Koch, Hans; Roquejoffre, Jean-Michel
    We investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds.