Vanishing-mass limit for locally constrained IMD problems

Let $p\in(1,\infty)$ and $t_{\max}>0$. For each natural number $n$, let $(\hat{k}_n,\hat{\mu}_n)\in L^\infty(\Omega;\mathbb{R}_+)^2$ solve the (vp-IMD) problem, respectively the (sp-IMD) problem, with reference modulus $E_{0,n}=E_0/n$ and the additional local constraint. The vanishing-mass conjecture. After passage to a subsequence,

$$(n\hat{k}_n,n\hat{\mu}_n)\rightharpoonup(\hat{k},\hat{\mu})$$

weakly in $L^p(\Omega;\mathbb{R}^2)$, where $(\hat{k},\hat{\mu})$ solves the corresponding unmodified (vp-IMD) problem, respectively the unmodified (sp-IMD) problem. This predicts that rescaling the moduli compensates for the vanishing reference modulus and removes the effect of the local constraint in the limit. The paper describes the claim as heuristic and leaves a corresponding delicate $\Gamma$-convergence analysis for future work.