Title page for ETD etd-11142005-143000


Type of Document Dissertation
Author Cazacu, Rodica
Author's Email Address rcazac1@lsu.edu
URN etd-11142005-143000
Title Quasicontinuous Derivatives and Viscosity Functions
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Jimmie Lawson Committee Chair
Charles Monlezun Committee Member
Daniel C. Cohen Committee Member
Frank Neubrander Committee Member
William Adkins Committee Member
Suresh Rai Dean's Representative
Keywords
  • bicontinuous lattice
  • quasicontinuous
  • viscosity solution
  • approximate function
  • domain theory
Date of Defense 2005-11-10
Availability unrestricted
Abstract
In this work we demonstrate how the continuous domain theory can be applied to the theory of nonlinear optimization, particularly to the theory of viscosity solutions. We consider finding the viscosity solution for the Hamilton-Jacobi equation H(x, y) = g(x), with continuous hamiltonian, but with possibly discontinuous right-hand side. We begin by finding a new function space Q(X,L), the space of equivalence classes of quasicontinuous functions from a locally compact set X to a bicontinuous lattice L and we will define on Q(X,L) the qo-topology, which is a variant of classical order topology defined on complete lattices. On this new function space we will show that there exist closed extensions of some differential operators, like the usual gradient and the operator defined by the continuous hamiltonian H. The domain of the closure of the corresponding operator will coincide with the set of viscosity solutions for the Hamilton-Jacobi equation when the hamiltonian is convex in the second argument.
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