#### Identifier

etd-11142005-143000

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### 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.

#### Date

2005

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Cazacu, Rodica, "Quasicontinuous derivatives and viscosity functions" (2005). *LSU Doctoral Dissertations*. 3533.

https://digitalcommons.lsu.edu/gradschool_dissertations/3533

#### Committee Chair

Jimmie Lawson