Available at: https://digitalcommons.calpoly.edu/theses/3190
Date of Award
12-2025
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Morgan Sherman
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
Many boundary value problems that arise in mathematical models have close connections to second order elliptic partial differential equations. This thesis introduces the idea of weak derivatives and Sobolev Spaces to generalize possible solutions. Using functional analysis centered around the Lax-Milgram theorem, we show the existence of these generalized solutions to boundary value problems including Laplace's Equation, 2nd order linear ODEs, and ultimately a general second order elliptic PDE. The work cumulates with recovering a number of central theorems of functional analysis in the context of Sobolev Spaces, creating a new perspective on the solvability of these boundary value problems.