The gradient $\nabla d$ might not be continuous across the interface where the speed changes. Is the contour integral of $\nabla d$ still equal to zero in this case?

Thank you for the reference and for the intuitive explanation!

To talk about $\Gamma$-convergence you need to define your functionals on the same space. If the natural spaces for $F_\varepsilon$ change with $\varepsilon$ you use $+\infty$ for elements outside your domain.

Have you tried separating into multiple simpler polynomials? You can easily see that parts of the terms come from the decomposition of $(x+a)(x+b)(x+c)(x+d)$... From where does this question come from?

I'm not sure, but can you reformulate this as a relaxation of a linear programming problem? The variables are $a_{ij}\in [0,1]$ (there is a link between points $i,j$ or not). The objective function is the sum of $a_{ij}l_{ij}$ (combination of the length of the segments). The constraint is $\sum a_{ij} = n$, have exactly $n$ segments. There might be some problems when two $l_{ij}$ are equal. Anyway, it's just an idea.

@username: it turns out it is possible to obtain an estimate by integrating by parts by hand. It is weird that I couldn't find any explicit references regarding this.

Do you have a precise reference for the initial result?

@username: Thank you for your suggestions. Indeed, in Grisvard's book, there is a section on a priori estimates, where such kind of estimates is presented. However, it is in a very general setting, and in the case of the Laplacian, I thought there may be a more direct reference where the computations are more straightforward. I will try your second suggestion.