Browsing by Subject "partial differential equations"
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Item Approximation of linear partial differential equations on spheres(Texas A&M University, 2004-09-30) Le Gia, Quoc ThongThe theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions.Item 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-MichelWe 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.