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What does minimizing the Dirichlet energy of a mapping $\Phi$ achieve intuitively? Roughly it is the integral (or sum, if discrete) of $|\nabla \Phi(\;)|^2 dV$, with $V$ the volume. So is it, in some sense, a minimally distorting (on average) mapping? Here is an example,
Figure 5
from

Yaron Lipman. "Construction of Injective Mappings Of Meshes." arXiv:1310.0955 [cs.CG].

But when I look at this figure and try to view it as minimally distorting, I don't see in what sense it minimally distorts—it greatly distorts! I feel like I am missing some basic intuition here. Any insights would be appreciated. Thanks!

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