Global ETD Search

1	Regularity Results for Potential Functions of the Optimal Transportation Problem on Spheres and Related Hessian Equations von Nessi, Gregory Thomas, greg.vonnessi@maths.anu.edu.au January 2008 (has links) In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form \begin{align} F\left[D^2u-A(\,\cdot\,,u,Du)\right] &= B(\,\cdot\,,u,Du),\quad\text{in}\ \Omega^-,\notag\\ T_u(\Omega^-) &= \Omega^+, \end{align} where $A$, $B$, $T_u$ are all prescribed; and $\Omega^-$ along with $\Omega^+$ are bounded in $\mathbb{R}^n$, smooth and satisfying notions of c-convexity and c^*-convexity relative to one another (see [MTW05] for definitions). In particular, the case where $F$ is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F, introducing some complications that are not present in the Optimal Transportation case.¶ In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres. partial differential equations PDE fully-nonlinear elliptic PDE Optimal Transportation geometric analysis calculus of variations
2	Fully linear elliptic equations and semilinear fractionnal elliptic equations Chen, Huyuan 10 January 2014 (has links) Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient, ... / This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term ... Propriété d'Hadamard Théorème de typr Liouville Solutions de ciscosité Laplacien fractionnaire Existence Unicité Comportement asymptotique Grandes solutions Mesures de Radon Mesure de Dirac Noyau de Green Capacité de Bessel Singularités isolées Solutions faibles Singularité faible Singularité forte Hadamard property Liouville type theorem Viscosity solution Fully nonlinear elliptic PDE Fractional Laplacian Existence Uniqueness Asymptotic behavior Blow-up solution Radon measure Dirac mass Green kernel Bessel capacities Isolated singularity Weak solution Weak singular solution Strong singular solution Propiedad de Hadamard Teorema del tipo de Liouville Soluciones Viscosas EDP elípticas completamente no lineales Laplaciano fraccionario Existencia, Unicidad Comportamiento asimptótico Soluciones blow-up Medida de Radon Masa de Dirac Núcleo de Green Capacidades de Bessel Singularidad aislada Soluciones débiles Soluciones singulares débiles Soluciones singulares fuertes

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Regularity Results for Potential Functions of the Optimal Transportation Problem on Spheres and Related Hessian Equations

Fully linear elliptic equations and semilinear fractionnal elliptic equations