Min–max minimal hypersurfaces with obstacle

Abstract

We study min–max theory for the area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen–Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restricts to a stable minimal hypersurface in the interior. Based on this, we show that for any admissible family of sweepouts $\Pi$ in a compact manifold with boundary, there always exists a closed $C^{1,1}$ hypersurface with codimension $\ge 7$ singular set in the interior and having mean curvature pointing outward along boundary realizing the width $\mathbf {L}(\Pi )$.

A Improved regularity for obstacle problems

The goal of this section is to prove Theorems 2.11 and 2.12. First recall that under local coordinate chart $(\mathcal {U}, x^i)$ of $\partial M$, if $u\in C^1(\mathcal {U}, (-r_M/2, r_M/2))$, then $\mathscr {H}^n(graph_{\partial M}(u))= \int _{\mathcal {U}} F(x, u, du)$, where $0< F(x, z, p)\in C^{\infty }(\mathcal {U}\times (-r_M, r_M)\times \mathbb {R}^n)$ satisfies

$$\begin{aligned} \partial ^2_{pp}F(x, z, p)^{ij}\xi _i\xi _j \ge \lambda (x,z, p) |\xi |^2 >0,\ \ \ \ \forall \xi \in \mathbb {R}^n \end{aligned}$$

(A.1)

$$\begin{aligned} \Vert F\Vert _{C^4(\mathcal {U}\times [-r_M/2, r_M/2]\times \mathbb {B}^n_R)} \le C(M, g, R). \end{aligned}$$

(A.2)

If $u\ge 0$ and $graph_{\partial M}(u)$ is constrained stationary in M, then

$$\begin{aligned} \begin{aligned} 0\le&\frac{d}{dt}\Big |_{t=0}\int _{\mathcal {U}} F(x, u+t\phi , du+td\phi ) \ dx \\ \le&\int _{\mathcal {U}} \partial _p F(x,u, du)\cdot d\phi + \partial _z F(x, u, du)\phi \ dx, \end{aligned} \end{aligned}$$

(A.3)

for every $\phi \in C_c^1(\mathcal {U})$ such that $0<t<<1$, $u+t\phi \ge 0$.

We shall deal with more general quasi-linear 2nd order elliptic operator of divergence form. For simplicity, write $B_r(x):= \mathbb {B}^n_r(x)\subset \mathbb {R}^n$, $B_r := B_r(0)$ and $\Omega _r:= B_r\times [-1, 1]\times B_1$. Let $\lambda , \Lambda >0$, $\alpha \in (0, 1)$ be constant, $A = A(x, z, p)\in C^3(B_2\times \mathbb {R}\times \mathbb {R}^n, \mathbb {R}^n)$, $B=B(x, z, p)\in C^2(B_2\times \mathbb {R}\times \mathbb {R}^n)$ satisfying

$$\begin{aligned} \begin{aligned} \partial _p A^{ij}|_{B_2\times [-1, 1]\times B_1} \cdot \xi _i\xi _j&\ge \lambda |\xi |^2,\ \ \ \ \forall \xi \in \mathbb {R}^n; \\ \Vert A\Vert _{C^3(\Omega _2)} + \Vert B\Vert _{C^2(\Omega _2)}&\le \Lambda . \end{aligned} \end{aligned}$$

(A.4)

Let $\mathscr {M}u := -div(A(x, u, \nabla u)) + B(x, u, \nabla u)$. Consider the variational inequality for a function $u\in C^1(B_2, \mathbb {R}_+)$,

$$\begin{aligned} \begin{aligned} \int _{B_2} A(x, u, \nabla u)\cdot \nabla \phi + B(x, u, \nabla u)\phi \ dx&\ge 0,\ \ \ \forall \phi \in C_c^1(B_2, \mathbb {R}_+); \\ \int _{B_2} A(x, u, \nabla u)\cdot \nabla \phi + B(x, u, \nabla u)\phi \ dx&= 0,\ \ \ \forall \phi \in C_c^1(\{u>0\}). \end{aligned} \end{aligned}$$

(A.5)

Clearly, (A.3) implies (A.5), where $A = \partial _p F$ and $B = \partial _z F$.

Theorem A.1

$\exists \delta _0 = \delta _0(\lambda , \Lambda , n, \alpha )\in (0, 1)$ such that if A, B satisfies (A.4), $0\le u\in C^{1,\alpha }_{loc}(B_2{\setminus } \{0\})\cap C^1(B_2)$ satisfies (A.5) and that $\Vert u\Vert _{C^1(B_2)}\le \delta _0$, then $u\in C^{1,1}_{loc}(B_2)$ and

$$\begin{aligned} \Vert u\Vert _{C^{1,1}(B_1)} \le C(\lambda , \Lambda , n, \alpha ). \end{aligned}$$

In view of the proof of Lemma 3.4, Theorem 2.11 follows from Theorem A.1.

The major effort is made to prove the following lemma,

Lemma A.2

There exists $\delta _0 = \delta _0(\lambda , \Lambda , n, \alpha )\in (0, 1)$, $\eta _0 = \eta _0(\lambda , \Lambda , n, \alpha )\in (0, \delta _0)$ s.t. if u be in Theorem A.1 and further satisfying

$$\begin{aligned} \Vert u\Vert _{C^{1, \alpha }(B_2)} \le 2\delta _0, \end{aligned}$$

(A.6)

and $\eta <\eta _0$ such that the elliptic coefficients satisfies

$$\begin{aligned} \Vert \partial _x A\Vert _{C^2(\Omega _2)} +\Vert \partial _z A\Vert _{C^2(\Omega _2)} + \Vert B\Vert _{C^2(\Omega _2)} \le \eta . \end{aligned}$$

(A.7)

Then $\Vert u\Vert _{C^{1,1}(B_1)} \le C(\lambda , \Lambda , n, \alpha )(\Vert u\Vert _{C^{1,\alpha }(B_2)} + \eta )$.

Proof of Theorem A.1

Observe that if $x_0\in B_1$, $r\in (0, 1)$, let $\tilde{u}(y):= u(x_0 + ry)/r$, then

$$\begin{aligned} \mathscr {M}u (x) = \frac{1}{r} [-div \tilde{A}(y, \tilde{u}(y), \nabla \tilde{u}(y)) + \tilde{B}(y, \tilde{u}(y), \nabla \tilde{u}(y))]_{y=(x-x_0)/r}, \end{aligned}$$

where

$$\begin{aligned} \tilde{A}(y, z, p)&:= A(x_0+ry, rz, p); \\ \tilde{B}(y, z, p)&:= rB(x_0+ry, rz, p). \end{aligned}$$

Hence, by Lemma A.2, for u satisfying the assumptions in Theorem A.1, $u\in C^{1,1}_{loc}(B_2{\setminus } \{0\})$. Thus, for a.e. $x\in B_2{\setminus } \{0\}$, $\mathscr {M}u (x) = B(x, 0, 0)$. Together with (A.5) and that $\Vert u\Vert _{C^1(B_2)} \le 1$, by a cutting off argument, $\mathscr {M}u (x) = B(x, 0, 0)\cdot \chi _{\{u=0\}}$ in the weak sense on $B_2$. Hence by standard elliptic estimate [17, Theorem 13.1], $\exists \beta =\beta (\lambda , \Lambda , n)\in (0 ,1)$ such that $\Vert u\Vert _{C^{1, \beta }(B_{3/2})}\le C(\lambda , \Lambda , n)$. Then repeat the rescaling argument above, by Lemma A.2, Theorem A.1 is proved. $\square$

Now we start to prove Lemma A.2.

Lemma A.3

$\exists \eta _1 = \eta _1(\lambda , \Lambda , n, \alpha )\in (0,1)$ such that if $\eta \le \eta _1$ and A, B satisfies (A.4) and (A.7), $\Vert \phi \Vert _{C^{1, \alpha }(\partial B_1)}\le \eta _1$. Then the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathscr {M}u = 0\ &{} \text { in }B_1 \\ u = \phi \ &{} \text { on }\partial B_1 \end{array}\right. } \end{aligned}$$

(A.8)

has a unique solution $u\in C^{1,\alpha }(Clos(B_1))$ satisfying

$$\begin{aligned} \Vert u\Vert _{C^{1, \alpha }}\le C(\lambda , \Lambda , n, \alpha )(\Vert \phi \Vert _{C^{1, \alpha }}+ \eta ). \end{aligned}$$

(A.9)

The uniqueness follows directly from comparison principle, see [17, Chapter 10]; We assert that the existence result does not follow directly from the classical Schauder estimate, since the latter requires boundary value to be $C^{2,\alpha }$. Instead, one need to work in the space $H_a^{(b)}$ introduced [14], where $a = 2+\alpha$, $b = 1+\alpha$. By [14, Theorem 5.1] and a fixed point argument, there exists $u\in H^{(b)}_a(B_1)$ satisfying (A.8) with uniformly bounded $H^{(b)}_a$-norm; and (A.9) follows from [14, Lemma 2.1].

Proof of Lemma A.2

Let $\delta _0, \eta _0\in (0, \eta _1)$ to be determined; u be in Lemma A.2.

Step 1 $\forall x_0\in B_{3/2}\cap \{u=0\}$, $\forall r\in (0, 1/2)$, let $v=v_{x_0, r}\in C^{1,\alpha }(B_r(x_0))$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathscr {M}v = 0\ &{} \text {in}B_r(x_0) \\ v = u\ &{} \text {on}\partial B_r(x_0) \end{array}\right. } \end{aligned}$$

given by Lemma A.3. Also, let $\bar{v}$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathscr {M}\bar{v} = 0\ &{} \text {in}B_r(x_0) \\ \bar{v} = 0\ &{} \text {on}\partial B_r(x_0). \end{array}\right. } \end{aligned}$$

Also note that $\Vert u\Vert _{C^1(B_r)}, \Vert v\Vert _{C^1(B_r)}, \Vert \bar{v}\Vert _{C^1(B_r)}\le C(\lambda , \Lambda , n, \alpha )(\eta _0 + \delta _0)$ and $\mathscr {M}u \ge 0$ in the weak sense. Hence, by taking $\eta _0, \delta _0<< 1$ and using the maximum principle [17, Theorem 10.10],

$$\begin{aligned} 0\le u(x)- v(x) \le C(\lambda , \Lambda , n, \alpha )\cdot \sup _{B_r(x)\cap \partial \{u>0\}} (u - v) \le C(\lambda , \Lambda , n, \alpha )\Vert v^-\Vert _{C^0(B_r(x_0))}, \end{aligned}$$

(A.10)

for every $x\in B_r(x_0)$; And $\bar{v}\le v$ on $B_r(x_0)$. Applying the classical $C^0$ estimate on $\tilde{v}(y):=\bar{v}(x_0+ry)/r$, one get

$$\begin{aligned} \Vert \bar{v}\Vert _{C^0(B_r(x_0))} \le C(\lambda , \Lambda , n, \alpha )r^2\Vert B(\cdot , 0, 0)\Vert _{C^0(B_r)}. \end{aligned}$$

(A.11)

Since $v-\bar{v}\ge 0$ on $B_r(x_0)$, $\mathscr {M}v = \mathscr {M}\bar{v} = 0$ and $v(x_0)\le u(x_0)=0$, by (A.4) and Harnack inequality,

$$\begin{aligned} \Vert v-\bar{v}\Vert _{C^0(B_{r/2}(x_0))} \le C(\lambda , \Lambda , n, \alpha )(v(x_0) - \bar{v}(x_0)) \le C(\lambda , \Lambda , n, \alpha )|\bar{v}(x_0)|. \end{aligned}$$

(A.12)

Combining (A.10), (A.11) and (A.12) one derive

$$\begin{aligned} 0\le u \le C(\lambda , \Lambda , n, \alpha )\Vert B(\cdot , 0, 0)\Vert _{C^0(B_r(x_0))}r^2\ \ \ \text { on }B_{r/2}(x_0). \end{aligned}$$

(A.13)

Step 2 $\forall x_0\in \{u=0\}$, $\nabla u(x_0) = 0$; $\forall x_0\in \{u>0\}\cap B_{3/2}$, if $l:=dist(x_0, \{u=0\})<1/8$, let $y_0\in \{u=0\}$ such that $|x_0 - y_0| = l$. Then by (A.13),

Hence by interior gradient estimate,

$$\begin{aligned} {\frac{1}{l}\Vert \nabla u\Vert _{C^0(B_{l/2}(x_0))} + \Vert \nabla ^2 u\Vert _{C^0(B_{l/2}(x_0))} \le C(\lambda , \Lambda , n, \alpha ). } \end{aligned}$$

(A.14)

Now that, $\forall x_1, x_2\in B_1$, $\rho :=|x_1-x_2|<1/20$, $l_i:=dist(x_i, \{u=0\})$, $l_1\le l_2$.

(1):

If $l_2 \ge 1/8$, then directly by gradient estimate

$$\begin{aligned} |\nabla u(x_1)- \nabla u(x_2)|\le \rho \sup _{B_{\rho }(x_2)}\Vert \nabla ^2 u\Vert \le C(\lambda , \Lambda , n, \alpha )|x_1-x_2|; \end{aligned}$$

(2):

If $2\rho \le l_2< 1/8$, then by (A.14),

$$\begin{aligned} |\nabla u(x_1)- \nabla u(x_2)|\le \rho \sup _{B_{l_2/2}(x_2)}\Vert \nabla ^2 u\Vert \le C(\lambda , \Lambda , n, \alpha )|x_1-x_2|; \end{aligned}$$

(3):

If $l_2\le 2\rho$, then also by (A.14),

$$\begin{aligned}&|\nabla u(x_1)-\nabla u(x_2)| \le |\nabla u(x_1)|+|\nabla u(x_2)|\\&\quad \le C(\lambda , \Lambda , n, \alpha )(l_1+l_2)\le C(\lambda , \Lambda , n, \alpha )|x_1-x_2|. \end{aligned}$$

This proves the $C^{1,1}$ bound of u. $\square$

Proof of Theorem 2.12

The regularity results in literature focus on minimizing boundaries; For general closed integral current, the strategy is to first decompose them into sum of boundary of Caccioppoli sets. One subtlety is that a priori the graphs may have incompatible orientations. We rule out this by choosing $U_y$ small enough.

Recall that $M^{n+1}\subset \mathbb {R}^L$ is a compact submanifold with boundary, extended to a closed manifold $\tilde{M}^{n+1}\subset \mathbb {R}^L$; $\tilde{U}\subset \tilde{M}$ is relatively open and $U:=\tilde{U}\cap M$. Suppose $U\cap \partial M \ne \emptyset$. $T\in \mathcal {Z}_n(U)$ minimize mass among $\{S\in \mathcal {Z}_n(U): spt(T-S)\subset \subset U\}$. Call such T constrained minimizing in U.

WLOG $U, \tilde{U}, V:=\tilde{U}{\setminus } U$ are contractible and $\mathbf {M}_U(T)>0$. By decomposition theorem [30, Chap 6,27.6], there’s a decreasing sequence of $\mathscr {H}^{n+1}$-measurable subsets $\{U_j\subset \tilde{U}\}_{j\in \mathbb {Z}}$ such that

$$\begin{aligned} T = \sum _{j\in \mathbb {Z}} \partial [U_j];\ \ \ \ \Vert T\Vert = \sum _{j\in \mathbb {Z}} \Vert \partial [U_j]\Vert . \end{aligned}$$

In particular, $\mathbf {M}_W(T) = \sum _{j\in \mathbb {Z}}\mathbf {M}_W(\partial [U_j])$ for all $W\subset \subset \tilde{U}$, $spt(\partial [U_j])\subset U$ and $T_j := \partial [U_j]$ are also constrained minimizers in U (along $\partial M$). Thus for each j, either $spt[U_j]$ or $spt([\tilde{U}]- [U_j])$ contains V. WLOG, $spt(\partial [U_0])\cap \partial M \ne \emptyset$ and $spt[U_0]\supset V$. Hence $\forall j\le 0$, $spt[U_j]\supset V$; and $\forall j>0$, either $spt[U_j]\supset V$ or $spt[U_j]\subset spt[U_0]{\setminus } V$.

Claim: $\forall \mathcal {K}\subset U\cap \partial M$ compact, $\exists \ r_1 = r_1(M, U, \mathcal {K})>0$ such that if $T_0 = \partial [U_0]$, $T_1 = \partial [U_1]$ are constrained minimizers in U and $U_0\supset V$, $U_1\subset U_0{\setminus } V$; $y\in \mathcal {K}\cap spt(T_0)$. Then $U_1\cap B_{r_1}(y)=\emptyset$.

Clearly, with this claim, for each $j\in \mathbb {Z}$, $T_j\llcorner B_{r_1}(y) = \partial [U_j']\llcorner B_{r_1}(y)$ is constrained minimizing in $B_{r_1}(y)$, where $U_j' := U_j\cap B_{r_1}(y)$ is either empty or containing V. Hence, by [24] and [33], $T_j\llcorner B_{r_1}(y)$ is $C^{1,\alpha }$ for some $\alpha \in (0, 1)$. Then by Theorem 2.11, T is $C^{1, 1}$ multiple graphs restricting to a smaller subset $U_{y}$. This proves (1) of Theorem 2.12; and (2) follows from unique continuation of minimal hypersurfaces.

Proof of the claim: By [24], for every $\delta \in (0,1), \exists \ r_2 = r_2(M, U, \mathcal {K}, \delta )>0$ and $u_0\in C^1(U\cap \partial M)$ such that $\forall y\in \mathcal {K}\cap spt(T_0)$,

$$\begin{aligned} T_0\llcorner B_{r_2}(y) = [graph_{\partial M}(u)]\llcorner B_{r_2}(y) \text {with} |\nabla u|\le \delta . \end{aligned}$$

(A.15)

Assuming the claim fails, then $\exists \ y_j\in \mathcal {K}\cap spt(T_0^{(j)})$, $T_i^{(j)} = \partial [U_i^{(j)}]$ being constrained minimizing boundary in U, where $i=0,1$, $j\ge 1$, $U_0^{(j)}\supset V$, $U_1^{(j)}\subset U_0^{(j)}{\setminus } V$ and $U_1^{(j)}\cap B_{1/j}(y_j)\ne \emptyset$. Then $spt(T_1^{(j)})\cap B_{1/j}(y_j)\ne \emptyset$.

Let $j\rightarrow \infty$, $y_j\rightarrow y_{\infty }\in \mathcal {K}$, and consider $(\eta _{y_j, 1/j})_{\sharp }T_1^{(j)}\rightarrow T_1^{(\infty )}$ being constrained minimizing in $T_{y_{\infty }}^+M \cong \mathbb {R}^{n+1}_+$. The convergent is both in current and in varifold sense. Therefore, by (A.15) and that $y_j\in \mathcal {K}\cap spt(T_0^{(j)}) \subset \partial M$, $U_1^{(j)}\subset U_0^{(j)}{\setminus } V$, we have $T_1^{(\infty )}=\partial [U_1^{(\infty )}]$ for some $spt(U_1^{(\infty )})\subset T_{y_\infty }\partial M$, which is $n+1$ negligible. In particular, $T_1^{(\infty )} = 0$; On the other hand, by $spt(T_1^{(j)})\cap B_{1/j}(y_j)\ne \emptyset$, Lemma 2.3 and Corollary 2.4, $T_1^{(\infty )}\ne 0$ and we get a contradiction. $\square$