# Geometric analysis: regularity theory for subelliptic PDEs and incompatible elasticity

Ricciotti, Diego (2018) Geometric analysis: regularity theory for subelliptic PDEs and incompatible elasticity. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## 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Status: Unpublished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:
Ricciotti, Diegodir17@pitt.edudir17@pitt.edu
ETD Committee:
Committee ChairManfredi, Juanmanfredi@pitt.edumanfredi@pitt.edu
Committee CoChairLewicka, Martalewicka@pitt.edulewicka@pitt.edu
Committee MemberHajlasz, Piotrhajlasz@pitt.eduhajlasz@pitt.edu
Committee MemberLeoni, Giovannigiovanni@andrew.cmu.edu
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