Dimension Reduction and Regularity of Perimeter Minimizers

Theorem 3.3.1

Suppose \(E\) is a perimeter minimizer in \(\mathbb{R}^{n+1}\). We denote \(V=|\partial^* E|\). Then, we have the following regularity result: \(\mathrm{sing}\|V\|=\emptyset\) if \(n\leq 6\), \(\mathrm{sing}\|V\|\) is discrete if \(n=7\), and \(\mathcal{H}^{n-7+\delta}(\mathrm{sing}\|V\|)=0\) for any \(\delta\gt{}0\) for \(n\geq 8\).

Theorem 3.3.2

Let \(\mathcal{V}\) be the collection of all the varifolds in the preceding theorem, i.e., the varifolds corresponding to the perimeter minimizers. Then,

\[ \operatorname{dim}(\mathrm{sing}(\|V\|\cap B_1))\leq n-7. \]

In particular, if \(n=7\), then \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete.

Proof. We denote \(F^l=\{ V \in \mathcal{V}: \mathcal{H}^l(\mathrm{sing}\cap B_1)\gt{}0 \}\).

Proposition 3.3.3

For each \(V \in F^l\), there exists \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\cap F^l\) for \(\mathcal{H}^l-\)a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\).

Proof. Recall that we actually have for \(\mathcal{H}^l\)-a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\), we have

\[ \limsup_{r \to 0} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_r(x))}{\omega_n r^l}\gt{}0. \]

This is equivalent to \(\mathcal{H}^l(A)\gt{}0 \equiv \mathcal{H}^l_\infty(A)\gt{}0\) for any \(A \subset \mathbb{R}^n\).

We choose \(r_i \to 0\) such that

\[ \lim_{i \to \infty} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_{r_i}(x))}{\omega_n r_i^l}\gt{}0. \]

By taking a subsequence, we can assume \((\eta_{x,r_i})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\).

If \(\mathcal{H}^l_\infty(\mathrm{sing}\,\|\boldsymbol{C}\|)=0\), then for any \(\varepsilon\gt{}0\), we can find a covering of \(\mathrm{sing}\,\|\boldsymbol{C}\|\) by balls \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) such that

\[ \sum_{j=1}^\infty s_j^l \lt{} \varepsilon. \]

Note that \(\mathrm{sing}\,\|\boldsymbol{C}\|\cap B_1(0)\) is compact, we know \(\mathrm{sing}\,\|(\eta_{x,r_i})_{\#}V\|\cap B_1(0)\) can also be covered by \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) for \(i\) large enough. Thus, we have

\[ \frac{\mathcal{H}^l_\infty(\mathrm{sing}\,\|V\|\cap B_{r_i}(x))}{r_i^l}\lt{}\varepsilon \]

for \(i\) large enough, which is a contradiction. ◻

Now we apply the above proposition iteratively to obtain a sequence of varifolds \(\{ V_k \}_{k=0}^{K}\) such that:

\(V_0=V\).
\(V_{k+1} \in \mathrm{VarTan}(V_k,x_k)\) for some \(x_k \in \mathrm{sing}(\|V_k\|)\cap B_1 \backslash \mathcal{S}(V_k)\).
\(\operatorname{dim}(\mathcal{S}(V_{k+1}))\gt{}\operatorname{dim}(\mathcal{S}(V_k))\) for each \(0\leq k \leq K-1\).
\(\mathcal{H}^l(\mathrm{sing}(\|V_k\|)\cap B_1)\gt{}0\) for each \(0\leq k \leq K\).
\(V_K=\boldsymbol{C}\times \mathbb{R}^{m}\) where \(\boldsymbol{C}\backslash \{0\}\) is a smooth immersed cone for some \(m\geq 0\).

In particular, the last two conditions imply \(m=l\). Now, \(\boldsymbol{C}\) becomes a stable cone with isolated singular point \(0\) of dimension \(n-l\). By the classification of the stable cones, we know \(n-l\geq 7\), and hence \(l\leq n-7\). This shows \(\operatorname{dim}(\mathrm{sing}\|V\|\cap B_1)\leq n-7\).

In the case \(n=7\), we have

\[ \mathcal{H}^\alpha(\mathrm{sing}\,\|V_K\|\cap B_1)=0 \]

for any \(\alpha\gt{}0\) by the above result.

If \(\mathrm{sing}(\|V\|\cap B_1)\) is not discrete, then we can find \(x_j \in \mathrm{sing}(\|V\|\cap B_1)\) such that \(x_j \to x_0 \in \mathrm{sing}(\|V\|\cap B_1)\). Now, up to a subsequence, we can assume \((\eta_{x_0,|x_j-x_0|})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)\) and we denote \(\xi = \lim_{j \to \infty} \frac{x_j-x_0}{|x_j-x_0|} \neq 0\). So \(\mathcal{S}(\boldsymbol{C})\) contains the line spanned by \(\xi\). In particular, \(\mathcal{H}^1(\mathrm{sing}(\|\boldsymbol{C}\|)\cap B_1)\gt{}0\) which is a contradiction. Hence, \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete. ◻

Dimension Reduction and Regularity of Perimeter Minimizers

Gaoming Wang

Assistant Professor