Partial Differential Equations on Low-Dimensional Structures

Łukasz Chomienia

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arXiv · 2404.10657

Partial Differential Equations on Low-Dimensional Structures

Łukasz Chomienia

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Abstract

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.

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Łukasz Chomienia. 2024-04-16. Partial Differential Equations on Low-Dimensional Structures. https://arxiv.org/abs/2404.10657

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