<p>Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(<i>u</i>) ≡ <i>u</i><sup>2</sup><sub>x</sub><i>u</i><sub>xx </sub>+ 2<i>u</i><sub>x</sub><i>u</i><sub>y</sub><i>u</i><sub>xy </sub>+<sub> </sub><i>u</i><sup>2</sup><sub>y</sub><i>u</i><sub>yy </sub>= 0. For numerical reasons ∆<i>q</i>(<i>u</i>) = div (|▼<i>u</i>|<i>q</i>▼<i>u</i>)<i> = </i>0, which (formally) approaches as ∆∞(<i>u</i>) = 0 as <i>q</i> → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(<i>u</i>) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:liu-3724 |
| Date | January 2005 |
| Creators | Hansson, Mattias |
| Publisher | Linköping University, Department of Mathematics, Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, text |
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