Ricciotti, Diego
(2018)
Geometric analysis: regularity theory for subelliptic PDEs and incompatible elasticity.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
This thesis is divided in two parts, which share a common theme of analysis in non-Euclidean spaces.
The first one focuses on regularity of weak solutions of the $p$-Laplace equation in the Heisenberg group. In particular, we give a proof of the fact that, for $p>4$, solutions assumed to be in the horizontal Sobolev space $HW^{1,p}$ (consisting of $L^p$ functions whose horizontal gradient is in $L^p$), possess H\"older continuous horizontal derivatives.
The argument is based on approximation via solutions of regularized problems: estimates independent of a non degeneracy parameter are obtained and passed to the limit. In particular, we show that the horizontal derivatives belong to a weighted De Giorgi space and then employ an alternative argument, not unlike the Euclidean case.
The second part deals with non-Euclidean elasticity. We study incompatibly prestrained thin plates characterized by a prescribed Riemannian metric on their reference configuration. We analyze scaling of the elastic energy $E^h$ of order higher than $2$ in plate's thickness $h$, i.e. $\inf h^{-\beta}E^h$ for $\beta>2$.
We find that, within this range, the only possible non trivial scaling is $\beta=4$. In this case we identify and study the $\Gamma$-limit functional, which consists of a von K\'arm\'an-like energy, given in terms of the first
order infinitesimal isometries and of the admissible
strains on the surface isometrically immersing the prestrain metric on the
midplate in $\mathbb{R}^3$.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
| Creators | Email | Pitt Username | ORCID  |
|---|
| Ricciotti, Diego | dir17@pitt.edu | dir17@pitt.edu | |
|
| ETD Committee: |
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| Date: |
27 September 2018 |
| Date Type: |
Publication |
| Defense Date: |
20 April 2018 |
| Approval Date: |
27 September 2018 |
| Submission Date: |
5 July 2018 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
90 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
p-Laplacian, Heisenberg group, Gamma-convergence |
| Date Deposited: |
27 Sep 2018 20:16 |
| Last Modified: |
27 Sep 2018 20:16 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/34653 |
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