Partial Differential Equations on Low-Dimensional Structures

This thesis pertains to the study of elliptic and parabolic partial differential equations on "thin" structures. The first main objective is to establish the strong and weak low-dimensional counterparts of the parabolic Neumann problem. The main technical result is proving the closedness of the low-dimensional second-order operator. To construct a semigroup, a variant of Magyar of the Hille-Yosida Theorem for non-invertible operators is adapted. An alternative direction of study is presented to extend the class of accessible initial data. Weak-type parabolic problems are defined, and the existence of solutions is obtained by the application of the Lions version of the Lax-Milgram Lemma. The second aspect of the thesis is to examine the higher regularity of weak solutions to low-dimensional elliptic problems. We prove that for any weak low-dimensional elliptic solution, some Cosserat vector field exists as a witness of the membership of the solution to the domain of the second-order operator.