Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer.

We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants $\Lambda$ and $r_0$ with $$0 \le \Lambda< \infty, \quad r_0>0$$such that $$P(E;B(x,r))\le P(F;B(x,r))+\Lambda|E\Delta F|$$whenever $E\Delta F \subset \subset B(x,r) \cap A$ and $r < r_0$.

In the proof, actually the term $\Lambda|E\Delta F|$ appears as a higher order perturbation of local perimeter minimality, thus it's not too difficult for one to guess the pattern of the proof. Basically the proof contains the following steps:

First, prove the weak regularity, i.e., the density estimate of points on the boundary of the minimizer. The proof is simple, by using a fairly straightforward comparison argument.

Second, prove the height bound theorem by considering the smallness of cylindrical excess. The proof basically uses the volume-area estimate several times, similar to De Giorgi's $L^2-L^{\infty}$ estimate in the proof of Holder regularity for elliptic equations with bounded coefficients of divergent form. The height bound theorem paves the way of Lipschitz approximation.

Then comes with the reverse Poincare Inequality, which says the cylindrical excess can be bounded by the cylindrical flatness. The proof is a little technical by constructing the cone-like comparison sets.

The next step is the harmonic approximation and "excess improvement by tilting" estimate. The reason to use the harmonic approximation is essentially because the area functional and the Dirichlet integral are close, and the improvement by tilting is because of the reverse Poincare inequality.

Then comes with the routine iteration technique to prove the Lipschitz regularity, together with Companato's criteria gives the $C^{1,\alpha}$ regularity of reduced boundary of the minimizer.

Finally, relates the problem to PDEs.

Regularity theorems for reduced boundaries can be proved under weaker minimality assumptions, but as far as I know, the minimizer has to be the "almost" perimeter minimizer, in some certain sense. Only in this case, some comparison argument and facts about distributional mean curvature can be used. Or intuitively, since we consider the regularity of the boundary, then it's "natural" to translate the problem into a certain kind of perimeter minimizer problem. Precisely, Tamanini's theorem shows that if $A \cap \partial ^*E$ is a $C^{1,\alpha}$-hypersurface for some $\alpha \in (0,1/2]$, then E is an almost perimeter minimizer in $A$.

Now my question is, given a geometric variational problem, sometimes I find it hard to relate the minimizer to be something like the perimeter minimizer. For example, under some conditions, it's not hard to prove that the existence of the minimizer of $$V(E)=\frac{\int_E g d \mathcal{L}^n}{H^{n-1}(E)}, E \subset \Omega$$where $\Omega$ is a bounded open set and $E$ is a set of finite perimeter. We know if $g=-1$, then the minimizer is the cheeger set, whose regularity is already known. But for general $g$, how to do the regularity provided that the minimizer exists? I cannot relate this problem to something like a perimeter minimizing problem. And what if for more general problems? For example, adding some more terms to the above functional. Can anyone give me some references to read or ideas to work on these kinds of problems? Maybe they are artificial, but I feel they are interesting, because it's hard for me to figure out a counterexample!

I'm a beginner of geometric measure theory, and any comments and suggestions would be highly appreciated. Thanks!


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