When is perimeter continuous under Hausdorff convergence?

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) ?

• In your problem how do the domains $D_n$ behave for large $n$? Do they converge to some $D$ as $n \to \infty$? Do they form a monotone sequence, perhaps? Oct 20, 2020 at 12:47
• @LeoMoos: Sorry, I forgot to mention that. The domains $D_n$ converge to some non degenerate domain $D$ in the Hausdorff metric. They do not necessarily form a monotone sequence. Oct 20, 2020 at 12:54
• Have you tried using the results from the compactness and regularity theory for sets with minimising properties? It seems to me that the limit should be regular away from a finite set of points; over the regular portions of the limit surface one ought to be able to apply an Allard regularity theorem for example to deduce the continuity you seek. (Note that this would also exclude the pathological cases you describe.) Oct 20, 2020 at 14:56
• @LeoMoos: I was hoping to find some results which prevent me from doing the proof from scratch. It is sure that the sets which interest me are regular (portions of circles/spheres) except some finite (and bounded) number of singular regions. I will look into your advice. Oct 21, 2020 at 12:10
• My point was that in the situation you describe (with sets that are 'minimal relative perimeter sets') the pathologies listed can (most likely) be avoided altogether, thereby precisely allowing you to avoid writing an argument from scratch. Oct 21, 2020 at 13:28

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So, as I understand it, you want to rule out "oscilatory" problems like this.

A sequence of $$C^\infty$$ regions that converge to the unit disk, but their perimeters converge to $$\infty$$.

If your $$\Omega_n$$ are $$C^2$$ regions, then curvature makes sense, so maybe a good condition will be a uniform bound on the curvature?

EDIT: this answers a different (or part of the) question, as Leo Moos remarked. To get an answer to the question, one would need to show that the $$\Omega_n$$ converge in the Hausdorff distance, and apply the answer to them instead of $$D_n$$.

With adequate assumptions ("$$\mu$$-reach" bounded below) similar to your intuition about possible failure cases, Theorem 4 in:

https://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/ccslt-scm-09/ccslt-scm-09.pdf

will guarantee that the perimeter of the $$r$$-neighborhood of $$D_n$$ will converge to the perimeter of the $$r$$-neighborhood of the limit.

Also, it shouldn't be too hard (assuming uniformly bounded total curvature) to show that the perimeters of $$r$$-neighborhoods of $$D_n$$ converge to the perimeters of $$D_n$$ as $$r$$ go to zero, uniformly in $$n$$.

It remains to to show that the limits can be "swapped". I believe that choosing $$r$$ to be a suitable function of the Hausdorff distance will work, thanks to the explicit form of the approximation error in Theorem 4 in the above paper.

• I think you're answering a slightly different question, no? In the setting described in the statement, one seeks the continuity of perimeter for sets $\Omega_n \subset D_n$, which are assumed to converge to some $\Omega \subset D$, not for the boundaries of the domains $D_n$ themselves. Perhaps I misunderstood the question... Oct 20, 2020 at 15:30
• Thank you for the reference. I will take a look to better understand what you mean. Uniformly bounded total curvature is a bit strong for what I need, since the sets I consider are only piecewise smooth, where the $C^1$ parts may meet at an angle (the number of angle points being bounded). Oct 20, 2020 at 15:57
• I meant that the integral of the absolute value of the curvature in uniformly bounded in $n$, which should be true here Oct 20, 2020 at 15:58
• @LeoMoos oh I see your point now. Edited my answer Oct 20, 2020 at 18:09

I found a paper which deals with the case I'm interested. It shows that for the particular case of minimal relative perimeter sets with given volume constraint the relative perimeter of the minimizers is continuous with respect to the Hausdorff metric when dealing with convex sets. The paper can be found here

The main idea is to exploit the fact that when convex sets $$C_i$$ converge to the convex set $$C$$ in the Hausdorff metric then there exist bilipschitz maps $$f: C_i \to C$$ with the property that the Lipschitz constants verify

$$\lim_{n \to \infty} \text{Lip}(f_i) = \lim_{n \to \infty} \text{Lip}(f_i^{-1})=1$$

Moreover, it is possible to give upper and lower bounds to perimeters/volumes of $$f_i(E_i)$$ and $$f_i^{-1}(E)$$ in terms of the Lipschitz constants and the perimeters/volumes of the sets $$E_i,E$$.

The lower semicontinuity is proved using a standard approach. The upper semicontinuity is proved by a contradiction argument.