Codenotti, Luca
(2018)
NUMERICAL IMPLEMENTATIONS
OF THEORETICAL RESULTS
IN PARTIAL DIFFERENTIAL EQUATIONS.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
This is the latest version of this item.
Abstract
In this work, we present two theoretical results in nonlinear partial differential equations and we
exploit both of them to produce novel visualizations of their solutions.
First, we show a proof of the existence and uniqueness of viscosity solutions to the p-Laplace
equation in the setting of the double obstacle problem. These solutions are built by adopting
the framework provided by so-called random tug-of-war games. Using the theoretical result, in
this context we employ a finite elements method to obtain visualizations of various approximate
solutions.
Second, we develop a proof of the density of C 1,α solutions to the Monge-Ampère equation in the
set of continuous functions. This proof was obtained in the framework provided by the technique of
convex integration. The proof is written with all due details, which allowus to give explicit bounds
for every involved quantity.
By means of numerical computations, we construct approximations of anomalous solutions to the
Monge-Ampère equation, suitably guided by the bounds obtained by our theoretical results.
Finally, we use these approximations to give a graphical description of anomalous solutions. The
visualizations and the numerical results provide insight into the approximating constructions.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
28 June 2018 |
| Date Type: |
Publication |
| Defense Date: |
12 December 2017 |
| Approval Date: |
28 June 2018 |
| Submission Date: |
11 April 2018 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
117 |
| 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: |
Double Obstacle Problem,
Monge Ampere |
| Date Deposited: |
28 Jun 2018 14:48 |
| Last Modified: |
28 Jun 2018 14:48 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/34501 |
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NUMERICAL IMPLEMENTATIONS
OF THEORETICAL RESULTS
IN PARTIAL DIFFERENTIAL EQUATIONS. (deposited 28 Jun 2018 14:48)
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