Optimality and threshold in higher-dimensional general-domain estimates

Determine whether the upper gradient estimates for finite-conductivity inclusions with low-conductivity imperfect interfaces in dimensions n e 3, without symmetry assumptions on the domains or boundary data, are optimal, and establish whether the threshold 2/(nmu) is the correct transition value.

Background

For finite-conductivity inclusions subject to low-conductivity-type imperfect interfaces, the paper proves general upper gradient bounds in dimensions n e 3 without symmetry assumptions. These bounds are uniformly bounded when the bonding parameter satisfies b3 c 2/(nmu), while for b3 e 2/(nmu) they grow at the rate predicted by the exponent balpha_0.

The paper establishes optimality for the corresponding estimates under symmetry assumptions and, in two dimensions, for general domains and boundary data. In dimensions n e 3 without symmetry, however, it leaves unresolved both whether the stated upper rates are sharp and whether the proposed threshold 2/(nmu) accurately describes the transition between bounded and blowing-up gradients.

References

However, it is not clear to us whether the estimates are optimal or whether the threshold is indeed 2/(n\mu) in this case.

— The conductivity problem with imperfect bonding interfaces and finite internal conductivities  (2610.05708 - Dong et al., 5 Oct 2026) in Section 1, Subsection "Main results," immediately following Theorem 1.3 (Theorem \ref{thm_upperbound_2}); reiterated in Section "Novelty, main difficulties, and organization"