Graduation Year

2024

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Thomas J. Bieske, Ph.D.

Committee Member

Andrei Barbos, Ph.D.

Committee Member

Razvan Teodorescu, Ph.D.

Committee Member

Sherwin Kouchekian, Ph.D.

Keywords

Grushin-Type Spaces, PDE, Sub-Riemannian Geometry, Viscosity Solutions, ∞-Laplacian, Elliptic Equations, Parabolic Equations

Abstract

This thesis poses the ∞-Laplace equation in Grushin-type spaces. Grushin-type spaces G are defined by the vector fields which serve as a basis for their tangent spaces; by weighting the canonical (Euclidean) directional vectors {∂/∂xi}ni=1 by functions ρi that obey certain technical assumptions, we produce a class of metric spaces in which certain directions may not be accessible at all points in the space. We prove the existence and uniqueness of viscosity solutions to both Dirichlet problems and Cauchy-Dirichlet problems involving the∞-Laplacian over bounded Grushin-type domains. The main tool in proving uniqueness of these solutions is a comparison principle for semilinear functions, which we obtain by exploiting the relationship between Euclidean and Grushin-type geometry. We also prove that solutions of certain Cauchy-Dirichlet problems converge to solutions of (time-stationary) Dirichlet problems as we permit the chronological variable t to tend toward ∞.

Download

Included in

Mathematics Commons

Share

COinS