Variational convergence of discrete minimal surfaces

Henrik Schumacher; Max Wardetzky

doi:10.1007/s00211-018-0993-z

Numerische Mathematik

January 2019, Volume 141, Issue 1, pp 173–213 | Cite as

Variational convergence of discrete minimal surfaces

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Henrik Schumacher
Max Wardetzky

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First Online: 03 September 2018

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Abstract

Building on and extending tools from variational analysis and relying on certain a priori assumptions, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth Douglas–Plateau problem under simplicial refinement. This convergence is with respect to a topology that is finer than the topology of uniform convergence of both positions and surface normals.

Mathematics Subject Classification

49Q05 53A10 58E12

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Notes

Acknowledgements

The authors would like to thank Heiko von der Mosel for discussions on the regularity of minimal surfaces.

Appendix A: Thickening robustness of Kuratowski limits

For the tickenings of set as introduced in Definition 5, we require the following result.

Lemma 17

Let (X, d) be a metric space, $r_n \ge 0$ with $\lim _{n\rightarrow \infty } r_n =0$ and $A_n\subset X$. Then one has

$$\begin{aligned} \mathop {{\hbox \mathrm{Li}}}\limits _{n\rightarrow \infty } \bar{B}(A_n, r_n) = \mathop {{\hbox \mathrm{Li}}}\limits _{n\rightarrow \infty } A_n \quad \text {and} \quad \mathop {{\hbox \mathrm{Ls}}}\limits _{n\rightarrow \infty } \bar{B}(A_n, r_n) = \mathop {{\hbox \mathrm{Ls}}}\limits _{n\rightarrow \infty } A_n. \end{aligned}$$

Proof

Monotonicity of the Kuratowski limits leads to

$$\begin{aligned} \mathop {\hbox {Li}}\limits _{n\rightarrow \infty } A_n \subset \mathop {\hbox {Li}}\limits _{n\rightarrow \infty } \bar{B}(A_n, r_n), \quad \text {and} \quad \mathop {\hbox {Ls}}\limits _{n\rightarrow \infty } A_n \subset \mathop {\hbox {Ls}}\limits _{n\rightarrow \infty } \bar{B}(A_n, r_n). \end{aligned}$$

Fix arbitrary $a \in \hbox {Ls}_{n\rightarrow \infty } \bar{B}(A_n, r_n)$, $\varepsilon >0$, and $N \in \mathbb {N}$ with $r_k < \tfrac{\varepsilon }{3}$ for all $k \ge N$. By definition of the Kuratowski limit superior, for each $n \ge N$ there is a $k \ge n$ and an $a_k \in B(a,\tfrac{\varepsilon }{3}) \cap \bar{B}(A_k,r_k)$. Now, choose $x_k \in A_k \cap B(a_k, 2 r_k)$ and observe $d(a,x_k) \le d(a,a_k) + d(a_k,x_k) <\tfrac{\varepsilon }{3} + 2 r_k \le \varepsilon $. Thus, we have $x_k \in B(a,\varepsilon ) \cap A_k \ne \emptyset $ for all such k. This shows $a \in \hbox {Ls}_{n\rightarrow \infty } A_n$.

Analogously, one shows $\hbox {Li}_{n\rightarrow \infty } A_n \supset \hbox {Li}_{n\rightarrow \infty } \bar{B}(A_n, r_n)$. $\square $

Appendix B: Estimates for Lipschitz immersions

In Lemma 21, we give a lower bound on the amount of $W^{1,\infty }_g$-perturbation that may be applied to a given Lipschitz immersion without leaving the space of Lipschitz immersions. Moreover, it allows us to to bound ${d_{{\text {Imm}}}}$ locally from above by the $W^{1,\infty }_g$-distance. Lemma 23 provides us with a reverse local bound.

In the following, we denote the operator norm of a linear operator A by $||{A}||$, while we use $|{A}|$ for the Frobenius norm. We will often write inner products with which these norms are defined as an index in order to resolve ambiguities.

Lemma 18

Fix b, $g \in {P}(V)$ and let $X \in T_b {P}(V) ={{\mathrm{Sym}}}(V)$ with $\left|X\right|_{g} < {{\text {e}}}^{-{d_P}(b,g)}$. Then $b+X$ is also contained in ${P}(V)$ and one has

$$\begin{aligned} {d_P}(b,b + X)&\le {{\text {e}}}^{{d_P}(g,b)} \, \left|X\right|_g. \end{aligned}$$

In particular, ${P}(V)$ is open in ${{\mathrm{Sym}}}(V)$.

Proof

Choose a g-orthonormal basis e of V and define $\mathbf {B} :={\mathbf {G}}_e(b)$, $\mathbf {X} :={\mathbf {G}}_e(X)$. Let $0<\lambda _1 \le \cdots \le \lambda _k$ be the eigenvalues of $\mathbf {B}$. Observe that

$$\begin{aligned} ||{\mathbf {B}^{-1}}||&= \lambda _1^{-1} \le \exp (|{\log (\lambda _1)}|) \le \exp (|{(\log (\lambda _1),\ldots , \log (\lambda _k))}|) = {{\text {e}}}^{{d_P}(b,g)}. \end{aligned}$$

The estimate

$$\begin{aligned} ||{\mathbf {B}^{-\frac{1}{2}}\mathbf {X}\mathbf {B}^{-\frac{1}{2}}}|| \le ||{\mathbf {B}^{-1}}|| \, ||{\mathbf {X}}|| \le ||{\mathbf {B}^{-1}}|| \, |{\mathbf {X}}| \le {{\text {e}}}^{{d_P}(b,g)} \cdot \left|X\right|_g<1 \end{aligned}$$

shows that $ \mathbf {B} + t \, \mathbf {X} = \mathbf {B}^{\frac{1}{2}}{}(\mathbf {I}_{n} + t\, \mathbf {B}^{-\frac{1}{2}}\mathbf {X}\mathbf {B}^{-\frac{1}{2}}){}\mathbf {B}^{\frac{1}{2}} $ is invertible for all $t \in [0,1]$. This implies that $\mathbf {B} + \mathbf {X}$ and thus $b + X$ are positive definite. Now, we have

$$\begin{aligned} {d_P}(b,b+X)&= |{\log {}(\mathbf {B}^{-\frac{1}{2}}( \mathbf {B} + \mathbf {X})\mathbf {B}^{-\frac{1}{2}} ){} }| \le |{\log {}( \mathbf {I}_k + \mathbf {B}^{-\frac{1}{2}}\mathbf {X}\mathbf {B}^{-\frac{1}{2}} ){} }| \\&\le |{\mathbf {B}^{-\frac{1}{2}}\mathbf {X}\mathbf {B}^{-\frac{1}{2}} }| =|{\mathbf {B}^{-1}\mathbf {X}}| \le |{\mathbf {B}^{-1}}| \, |{\mathbf {X}}| \le {{\text {e}}}^{{d_P}(g,b)} \, \left|X\right|_g, \end{aligned}$$

from which the stated estimate follows. $\square $

Lemma 19

Let $(V_i,g_i)$ be finite-dimensional Euclidean spaces for $i \in \{1,2\}$ and let $A \in {{\text {Mon}}}(V_1;V_2)$ be injective and let $B \in {{\mathrm{Hom}}}(V_1;V_2)$ with $\left|A-B\right|_{g_1,g_2} < \frac{1}{3} \exp (-\frac{3}{2} \ell )$, where $\ell :={d_P}(g_1,A^\#g_2)$. Then also B is injective and one has the estimate

$$\begin{aligned} {d_P}{}(A^\#g_2,B^\#g_2){} \le \tfrac{7}{3} \, {{\text {e}}}^{\frac{3}{2}\ell } \left|A-B\right|_{g_1,g_2}. \end{aligned}$$

Proof

We use the preceding lemma with $g=g_1$, $b:=A^\#g_2$, and $X:=B^\#g_2 - A^\#g_2$. Choose $g_i$-orthonormal bases $e_i$ of $V_i$ for $i \in \{1,2\}$ and write $\mathbf {X} :={\mathbf {G}}_{e_1}(X)$. Let $\mathbf {A}$ and $\mathbf {B}$ be the matrix representations of A and B, respectively, with respect to these chosen bases. Since $\mathbf {X} = \mathbf {A}^{\mathsf {T}}(\mathbf {B}-\mathbf {A}) + (\mathbf {B}-\mathbf {A})^{\mathsf {T}}\mathbf {A} + (\mathbf {B}-\mathbf {A})^{\mathsf {T}}\mathbf (\mathbf {B}-\mathbf {A})$ one obtains

$$\begin{aligned} \left|X\right|_g = |{\mathbf {X}}|&\le 2 ||{\mathbf {A}^{\mathsf {T}}\mathbf {A}}||^\frac{1}{2} |{\mathbf {B}-\mathbf {A}}|+ |{\mathbf {B}-\mathbf {A}}|^2 \le 2 \, {{\text {e}}}^{\frac{1}{2}\ell } \cdot \left|A-B\right|_{g_1,g_2} + \left|A-B\right|_{g_1,g_2}^2 \\&< {}(2 \, {{\text {e}}}^{\frac{1}{2}\ell } + \tfrac{1}{3} {{\text {e}}}^{-\frac{3}{2} \ell }){} \cdot \left|A-B\right|_{g_1,g_2} < {{\text {e}}}^{-\ell }, \end{aligned}$$

whenever $\left|A-B\right|_{g_1,g_2} < \frac{1}{3} {{\text {e}}}^{-\frac{3}{2} \ell }$. Finally, one has

$$\begin{aligned} {d_P}(b,b+X) \le {{\text {e}}}^{\ell } \, \left|X\right|_g \le {}(2 \, {{\text {e}}}^{\frac{3}{2}\ell } + \tfrac{1}{3} {{\text {e}}}^{-\frac{1}{2} \ell }){} \cdot \left|A-B\right|_{g_1,g_2} \le \tfrac{7}{3} \, {{\text {e}}}^{\frac{3}{2}\ell } \left|A-B\right|_{g_1,g_2} \end{aligned}$$

by Lemma 18. $\square $

Lemma 20

Let $(V_i,g_i)$ be finite-dimensional Euclidean spaces for $i \in \{1,2\}$, let ${d_{G_{1}}}$ be defined as in Example 5, and let $A \in {{\text {Mon}}}(V_1;V_2)$ be injective and let $B \in {{\mathrm{Hom}}}(V_1;V_2)$ with $\left|A-B\right|_{g_1,g_2} < \frac{1}{2} ||{A^{\dagger _{g_1,g_2}}}||_{g_2,g_1}^{-1}$. Then also B is injective and one has the estimate

$$\begin{aligned} {d_\infty }( {G_{1}}(A),{G_{1}}(B)) \le \uppi \, \sqrt{24} \, ||{A^{\dagger _{g_1,g_2}}}||_{g_2,g_1} \, |{A-B}|_{g_1,g_2}. \end{aligned}$$

Proof

For $u \in V_1'{\setminus } \{0\}$ and $U \in {{\mathrm{Hom}}}(V_1;V_2)$ with $\left|U\right| < \frac{1}{2}||{A^{\dagger }}||^{-1}$ define the function

$$\begin{aligned} F_u(U) :=\frac{\langle {A \,u , (A+U)\,u}\rangle }{\left|A \,u \right| \left|(A+U)\,u\right|}. \end{aligned}$$

This way, one has

$$\begin{aligned} {d_\infty }( {G_{1}}(A),{G_{1}}(A+U)) = \sup _{u \in V_1'{\setminus } \{0\}} \arccos (F_u(U))&\le \uppi \sup _{u \in V_1'{\setminus } \{0\}} \sqrt{2-2 F_u(U)} . \end{aligned}$$

We use a first order Taylor expansion of $F_u$ in order to bound $\sqrt{2-2 F_u(U)}$ from above. A direct computation shows that

$$\begin{aligned} F_u(0)=1, \quad DF_u(0)=0, \quad \text {and} \quad |{D^2 F_u(U)\, (V,V)}|\le 24 \, |{V}|^2 ||{A^\dagger }||^2 \end{aligned}$$

hold for all U, $V \in {{\mathrm{Hom}}}(V_1;V_2)$ with $|{U}| < \frac{1}{2} ||{A^\dagger }||$. Taylor’s theorem implies the existence of some $X \in {{\mathrm{Hom}}}(V_1;V_2)$ with $\left|X\right| \le \left|U\right|$ such that

$$\begin{aligned} F_u(U) = F_u(0) + DF_u(0) \, U + \frac{1}{2} D^2 F_u(X)\,(U,U) = 1 + \frac{1}{2} D^2 F_u(X)\,(U,U). \end{aligned}$$

Thus, we obtain the bound

$$\begin{aligned} \arccos (F_u(U)) \le \uppi \sqrt{2-2 F_u(U)}&= \uppi \sqrt{-D^2 F_u(X) \, (U,U)} \le \uppi \, \sqrt{24} \,||{A^\dagger }||\, \left|U\right|.\quad \end{aligned}$$

$\square $

Lemma 21

Let $f \in {{\text {Imm}}}(\varSigma ;{\mathbb {R}}^m)$, Open image in new window

, and $h \in W^{1,\infty }(\varSigma ;{\mathbb {R}}^m)$ with $ ||{f-h}||_{W^{1,\infty }_g} <\frac{1}{3} \exp (-\tfrac{3}{2} \ell ), $ where Open image in new window

. Then also h is a Lipschitz immersion and one has

$$\begin{aligned} {d_{{\text {Imm}}}}(f,h) \le 18 \, \exp (\tfrac{3}{2} \ell ) \, ||{f-h}||_{W^{1,\infty }_g}. \end{aligned}$$

In particular, ${{\text {Imm}}}(\varSigma ;{\mathbb {R}}^m)$ is an open subset in $W^{1,\infty }_g(\varSigma ;{\mathbb {R}}^m)$.

Proof

By Lemma 19, one has ${{\text {d}}}h|_x \in {{\text {Mon}}}(T_x\varSigma ;{\mathbb {R}}^m)$ for almost all $x \in \varSigma $. Lemmas 19 and 20 together imply

$$\begin{aligned} {d_{{\text {Imm}}}}(f,h)&\le \tfrac{7}{3} \, {{\text {e}}}^{\frac{3}{2}\ell } |{{{\text {d}}}f-{{\text {d}}}h}|_{L^\infty _g} + \uppi \, \sqrt{24} \, {}\big ( \mathop {\hbox {ess sup}}\limits _{x \in \varSigma } \,||{({{\text {d}}}f_x)^{\dagger _{g,g_0}}}|| \big ){}\, |{{{\text {d}}}f-{{\text {d}}}h}|_{L^\infty _g}. \end{aligned}$$

Now, $||{({{\text {d}}}f|_x)^{\dagger _{g,g_0}}}|| \le {{\text {e}}}^{\frac{1}{2}\ell } \le {{\text {e}}}^{\frac{3}{2}\ell }$ and $\frac{7}{3} + \uppi \sqrt{24} \le 18$ complete the proof. $\square $

Lemma 22

Let $(V_i,g_i)$ be finite-dimensional Euclidean spaces for $i\in \{1,2\}$ and let A, $B \in {{\text {Mon}}}(V_1;V_2)$ with ${d_P}(A^\#g_2,B^\#g_2) <\frac{5}{2}$. Then one has the estimate

$$\begin{aligned} ||{A-B}||_{g_1,g_2} \le \exp ({d_P}(g_1,A^\#g_2))\, {}\Big ( {d_{G_{1}}}({G_{1}}(A),{G_{1}}(B)) + {d_P}(A^\#g_2,B^\#g_2)\Big ){}. \end{aligned}$$

Proof

Let $u \in V_1{\setminus }\{0\}$. Let v and $w \in V_2$ be the unique unit vectors in the rays ${G_{1}}(A)([u])$ and ${G_{1}}(B)([u])$, respectively. Let $\lambda :=|{A \, u}|_{g_2}$ and $\mu :=|{B \, u}|_{g_2}$. With the triangle inequality, we obtain

$$\begin{aligned} |{(A-B)\,u}|_{g_2}&= |{A \,u - B\, u}|_{g_2} = |{\lambda \, v - \mu \, w}|_{g_2} \le |{\lambda \, v - \lambda \, w}|_{g_2} + |{\lambda \, w - \mu \, w}|_{g_2} \nonumber \\&\le \lambda |{v - w}|_{g_2} + |{\lambda -\mu }| \le \lambda \, {}\left( 2 \sin ( \tfrac{1}{2} \measuredangle (v,w)) + |{1-\tfrac{\mu }{\lambda } }|\right) {} \nonumber \\&\le ||{A}||_{g_1,g_2} \left|u\right|_{g_1}\, {}\Big ( {d_{G_{1}}}({G_{1}}(A),{G_{1}}(B)) + |{1-\tfrac{\mu }{\lambda } }| \Big ){}. \end{aligned}$$

(22)

Next, we choose an orthonormal basis e of $(V_1,A^\#g_1)$ and denote the eigenvalues of ${\mathbf {G}}_e(B)$ by $0<\lambda _1< \cdots < \lambda _ k \le \exp ({d_P}(A^\#g_2,B^\#g_2))$. One has $\sqrt{\lambda _1} \le \frac{\mu }{\lambda } \le \sqrt{\lambda _k}$, thus $ |{1-\tfrac{\mu }{\lambda } }| \le \max _{i=1,\ldots ,k} |{\sqrt{\lambda _i} -1}| $. From the inequality $|{\sqrt{t}-1}| \le |{\log (t)}|$ for all $t \in ]0,\exp (5/2)[$, it follows that $|{1-\tfrac{\mu }{\lambda } }| \le {d_P}(A^\#g_2,B^\#g_2)$. Substitution of this along with the bound $||{A}||_{g_1,g_2} \le \exp ({d_P}(g_1,A^\#g_2))$ into (22) and taking the supremum over all nonzero vectors u leads to the desired result. $\square $

This leads us directly to the following lemma.

Lemma 23

Let $(\varSigma ,g)$ be a compact Riemannian manifold with boundary and let f, $h \in {{\text {Imm}}}(\varSigma ;{\mathbb {R}}^m)$ be Lipschitz immersions with Open image in new window

. Then one has the estimate

Open image in new window

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Authors and Affiliations

Henrik Schumacher
- 1
Email author
Max Wardetzky
- 2

1.Institute for MathematicsRWTH Aachen UniversityAachenGermany
2.Institute for Numerical and Applied MathematicsGeorg-August-University GöttingenGöttingenGermany

Variational convergence of discrete minimal surfaces

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