# Browsing by Subject "Fully nonlinear equations"

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Item Fully nonlinear equations with applications to grad equations in plasma physics ; Interaction between a one phase free boundary problem and an obstacle problem ; Optimal trace Sobolev inequalities(2022-08-05) Tomasetti, Ignacio; Caffarelli, Luis A.; Maggi, Francesco, 1978-; Patrizi, Stefania; Stinga, Pablo RShow more In this thesis we address three different problems. First, we prove existence and regularity for a fully nonlinear and nonlocal equation which arises in plasma physics. This is a generalization of Grad Equations which model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a W [superscript 2,p]-viscosity solution and regularity up to C [superscript 1, alpha] [Ω overlined] for any α<1. Then we elaborate in how to improve this regularity near the boundary. The main ingredient to study is the nonlocality due to the presence of the measure of the superlevel sets in the equations. Second, we address a problem which models a reaction-diffusion process. Existence is proved, and the solution solves a one phase free boundary problem in the lower half space [doublestruck R] [superscript 3 underscored]. Its trace in [doublestruck R superscript 2] solves an obstacle problem for a given obstacle. We study the exchange between the diffusion in the horizontal plane and the lower half space. Third, we work with a family of variational problems with critical volume and trace constraints. This arises from the study of "best p-Sobolev inequalities" for n≥2 and p [is an element of](1,n). We extend the analysis from [MV05] and [MN17] for an open set Ω [is a proper subset of] [doublestruck R superscript n]. We prove existence of minimizers for Ω bounded and n>2p, and existence of generalized minimizers for n>p. We also establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" from [MV05].Show more Item Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones(2013-05) Chang Lara, Hector Andres; Caffarelli, Luis A.Show more On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary.Show more Item Regularity of elliptic transmission problems and a new family of integro-differential operators related to the Monge-Ampère equation(2022-07-12) Soria-Carro, María; Caffarelli, Luis A.; Stinga, Pablo Raúl; Gamba, Irene M; Patrizi, Stefania; Vasseur, Alexis FShow more This dissertation is divided into two main topics. First, we study transmission problems for elliptic equations, both linear and nonlinear, and prove existence, uniqueness, and optimal regularity of solutions. In our first work, we consider a problem for harmonic functions and use geometric techniques. Our second work considers viscosity solutions to fully nonlinear transmission problems. Given the nonlinear nature of these equations, our arguments are based on perturbation methods and comparison principles. The second topic is related to nonlocal Monge-Ampère equations. We define a new family of integro-differential equations arising from geometric considerations and study some of their properties. Furthermore, we consider a Poisson problem in the full space and prove existence, uniqueness, and C¹,¹ regularity of solutions. For this problem, we use tools from convex analysis and symmetrization.Show more Item The two membranes problem for fully nonlinear local and nonlocal operators(2019-02-11) Vivas, Hernán Agustín; Caffarelli, Luis A.; Maggi, Francesco; Vasseur, Alexis F; Arapostathis, AristotleShow more We study the Two Membranes Problem for fully nonlinear operators both in the local (second order) and nonlocal setting. The problem arises when studying a "bid an ask" model in mathematical finance where some asset has a price that varies randomly and a buyer and a seller have to agree on a price for a transaction to take place. The local/nolocal character of the problem, as well as the form of the operators considered, come precisely from the nature of the process. We give a mathematical formulation for the problem and prove existence of solutions in the viscosity sense via a penalization method. In the second order case we show the optimal C [superscript 1,1] regularity of solutions and provide an example showing that no regularity of the free boundary is expected to hold in general. In the nonlocal case we get C²[superscript s] regularity for solutions (2s − ε for any ε > 0 if s = 1/2). In order to achieve that, we prove regularity estimates for fully nonlinear nonlocal equations with bounded right hand side, a result that has interest on its own and is obtained combining a blow up argument with an appropriate Liouville-type theoremShow more