Trace/extension operators in rough domains and applications to partial differential equations
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Trace and extension theory lay the foundation for solving a plethora of boundary value problems. In developing this theory, one typically needs well-behaved extension operators from a specified domain to the entire Euclidean space. Historically, three extension operators have developed much of the theory in the setting of Lipschitz domains (and rougher domains); those due to A.P. Calderon, E.M. Stein, and P.W. Jones. In this dissertation, we generalize Stein's extension operator to weighted Sobolev spaces and Jones' extension operator to domains with partially vanishing traces. We then develop a rich trace/extension theory as a tool in solving a Poisson boundary value problem with Dirichlet boundary condition where the differential operator in question is of second order in divergence form with bounded coefficients satisfying the Legendre-Hadamard ellipticity condition.