Optimisation of a 1D family of Polyconvex Functionals

The following material is designed as an extension to the poster presented at Surrey & Oxford in 2023.

Additional Solutions to the PDI

Here we show some additional solutions to the PDI:

\[\begin{aligned} \nabla u \in \text{O}(2) \qquad u \in W^{1,2}_0 \left( Q; \mathbb{R}^2 \right) \end{aligned}\]

On the left we plot the original solution. In the middle we plot additional maps that, when composed with the original solution, preserve the PDI and boundary condition. The animations runs through several examples of these maps. On the right we then plot the result of this composition.

References

[1] Jonathan Bevan, On double-covering stationary points of a constrained Dirichlet energy, Annales de l’Institut Henri Poincarè C, Analyse non linèaire 31 (2014), no. 2, 391–411.

[2] Arrigo Cellina and Stefania Perrotta, On a problem of potential wells, J. Convex Anal 2 (1995), no. 1-2, 103–115.

[3] Bernard Dacorogna and Paolo Marcellini, Implicit partial differential equations, Progress in Nonlinear Partial Differential Equations and Their Applications, no. v. 37, Birkhäuser, Boston, 1999.

[4] Stefan Müller, Variational models for microstructure and phase transitions, vol. 1713, pp. 85–210, Springer Berlin Heidelberg, Berlin, Heidelberg, 1999.

[5] Vladimír Šverák, On the Problem of Two Wells, Microstructure and Phase Transition (Avner Friedman, Willard Miller, David Kinderlehrer, Richard James, Mitchell Luskin, and Jerry L. Ericksen, eds.), vol. 54, Springer New York, New York, NY, 1993, pp. 183–189.