# Browsing by Subject "Viscosity solutions"

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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 Several regularity results for nonlocal elliptic equations(2017-05) Yu, Hui, Ph. D.; Caffarelli, Luis A.; Figalli, Alessio; Jin, Tianling; Pavlovic, Natasa; Vasseur, Alexis FShow more Nonlocal elliptic equations have long been used by physicists and engineers to model diffusion processes involving jumps. Apart from several works from a probabilistic view, there had not been much development concerning their mathematical properties until the fundamental works of Caffarelli and Silvestre. Here we establish several results concerning the regularity of viscosity solutions to nonlocal elliptic equations. In particular, we show the existence of smooth solutions to two class of nonlocal fully nonlinear elliptic equations, an integrability estimate for the fractional order Hessian of solutions to nonlocal equations, as well as a theory of at solutions.Show more Item Singular limits of reaction diffusion equations of KPP type in an infinite cylinder(2007) Carreón, Fernando; Souganidis, TakisShow more In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case.Show more Item Uniqueness and existence results on viscosity solutions of some free boundary problems(2002-08) Kim, Christina; Souganidis, PanagiotisShow more