It is known that the perimeter is lower semicontinuous for the convergence of sets. Two variants are widely known:

(Golab's theorem) in $\Bbb{R}^2$ if the sets $\Omega_n$ converge to $\Omega$ in the Hausdorff metric then $\mathcal H^1(\partial \Omega) \leq \liminf \mathcal H^1(\partial E_n)$.

in general, when working with finite perimeter sets using total variation, if $\chi_{\Omega_n}$ converges to $\chi_\Omega$ in $L^1$ (convergence of characteristic functions) then again $\liminf Per(\Omega_n) \geq Per(\Omega)$.

These work very nice when dealing with minimization problems. When dealing with maximizing sequences, however continuity is necessary. This is known to be true in the convex case, i.e. if a sequence of convex sets $\Omega_n$ converge in the Hausdorff metric to $\Omega$ (with non-void interior) then the perimeters converge.

In the problem which interests me $\Omega_n$ are minimal relative perimeter sets inside some domains $D_n$ which may be considered convex. This means that they have a boundary which is piecewise $C^1$ with smooth parts having constant curvature. Moreover, the arcs meet at prescribed angles ($\pi/2$ with the boundary of $D_n$, $2\pi/3$ with one another). Such sets are not necessarily convex, but they are not at all arbitrary. Also, the domains $D_n$ converge in the Hausdorff metric to some domain $D$ (for simplicity assume $D_n$ and $D$ are convex and non degenerate, eventually with fixed volume). My guess is that we should have continuity of the perimeters in this case, but I failed to find results which help me conclude that. Therefore here are my questions (concerning sets which converge in the Hausdorff metric):

Are there any other

**pathological cases**where the perimeter is not continuous apart from cases where two parts of the boundary collapse or when an oscillatory boundary converges to some smoother limit (like zig-zags converging to a segment) ? (as said before, it is possible to assume that the sets are piecewise $C^1$)Do you know any concrete results where hypotheses under which the perimeter is continuous are discussed (apart from convexity) ?