$C^{1,\alpha}$ $h$-principle for von K\'arm\'an constraints

TL;DR

This paper establishes a $C^{1,eta}$ convex integration method for a specific nonlinear PDE system related to von Kármán constraints, leveraging connections to isometric immersion problems in two dimensions.

Contribution

It introduces a new convex integration approach for the von Kármán system by linking it to isometric immersion theory, providing a simplified construction method.

Findings

Constructs $C^{1,eta}$ solutions for the von Kármán system.

Connects the PDE system to isometric immersion problems.

Simplifies the convex integration process for these constraints.

Abstract

Exploiting some connections between the system $\nabla v\otimes\nabla v + 2$ sym $\nabla w = A$ and the isometric immersion problem in two dimensions, we provide a simple construction of $C^{1,\alpha}$ convex integration solutions for the former from the corresponding result for the latter.

Equations

$$(\nabla u{)}^T(\nabla u)=g.$$

$$\nabla v\otimes \nabla v+2\mathrm{sym}\nabla w=A$$

$$0<\alpha <\min \left\{\frac{1}{7},\frac{\beta }{2}\right\}.$$

$$\Vert \nabla \overline{v}-\nabla v{\Vert }_{C^0(\Omega )}\le C\Vert \nabla v\otimes \nabla v+2\mathrm{sym}\nabla w-A{\Vert }_{C^0(\Omega )}^{1/2}$$

$$\begin{array}{rl}\Vert \nabla \overline{w}-\nabla w{\Vert }_{L^p(\Omega )}& \le C\Vert \nabla v{\Vert }_{C^0(\Omega )}\Vert \nabla v\otimes \nabla v+2\mathrm{sym}\nabla w-A{\Vert }_{C^0(\Omega )}^{1/2}\\ & +\Vert \nabla v\otimes \nabla v+2\mathrm{sym}\nabla w-A{\Vert }_{C^0(\Omega )},\end{array}$$

$$2\mathrm{sym}\nabla \overline{w}+\nabla \overline{v}\otimes \nabla \overline{v}=A.$$

$$(u^{\ast }e{)}_{ij}={\partial }_{i}u\cdot {\partial }_{j}u.$$

$$|u(x)-u(y)|\le C|x-y{|}^{\beta }\text{mbox}forallx,y\in \Omega .$$

$$0<\alpha<\min\Big{\{}\frac{1}{1+n(n+1)},\frac{\beta}{2}\Big{\}},$$

$$\Vert g-g_0{\Vert }_0\le r$$

$$[g{]}_{\beta }\le {\theta }^2$$

$$\Vert u^{\ast }e-g{\Vert }_0\le {\delta }^2$$

$$\Vert {\nabla }^{2}u{\Vert }_0\le \mu ,$$

$$\Vert \nabla \overline{u}-\nabla u{\Vert }_0\le C\delta \text{mbox}and[\nabla \overline{u}-\nabla u{]}_{\alpha }\le C{\mu }^{\alpha }{\delta }^{1-\alpha }.$$

$$\Vert \nabla \overline{v}-\nabla v{\Vert }_0$$

$$\Vert \nabla \tilde{w}{\Vert }_{L^p}$$

$$+\Vert \nabla v\otimes \nabla v-\tilde{A}{\Vert }_{C^0(\Omega )}$$

$$t_0^2=\frac{r}{\Vert A{\Vert }_0}.$$

$$u_t(x)=\left(\begin{array}{c}x\\ tv(x)\end{array}\right)$$

$$u_t^{\ast }e-g_t=t^2(\nabla v\otimes \nabla v-A).$$

$${\delta }_t^{\beta -2}{\mu }_t^{-\beta }{\theta }_t^2$$

$$\Vert \nabla {\overline{u}}_t-\nabla u_t{\Vert }_0\le CDt\text{mbox}and[\nabla {\overline{u}}_t-\nabla u_t{]}_{\alpha }$$

$${\overline{u}}_t=\left(\begin{array}{c}{\overline{\Phi }}_t\\ t{\overline{v}}_t\end{array}\right).$$

$$\begin{array}{rl}[\nabla {\overline{v}}_t-\nabla v{]}_{\alpha }& \le C{M}^{\alpha }{D}^{1-\alpha }\\ \Vert \nabla {\overline{v}}_t-\nabla v{\Vert }_0& \le CD\end{array}$$

$$(\nabla {\overline{\Phi }}_t{)}^T(\nabla {\overline{\Phi }}_t)=I+t^2\left(A-\nabla {\overline{v}}_t\otimes \nabla {\overline{v}}_t\right).$$

$$\Vert \sqrt{\nabla {\overline{\Phi }}_t^T\nabla {\overline{\Phi }}_t}-I{\Vert }_{L^p}\le \Vert \nabla {\overline{\Phi }}_t^T\nabla {\overline{\Phi }}_t-I{\Vert }_{L^p}\le t^2{\Vert A-\nabla {\overline{v}}_t\otimes \nabla {\overline{v}}_t\Vert }_{L^p}.$$

$${\mathrm{dist}}_{SO(2)}(\nabla {\overline{\Phi }}_t)=\left|\sqrt{(\nabla {\overline{\Phi }}_t{)}^T(\nabla {\overline{\Phi }}_t)}-I\right|.$$

$$\Vert \nabla {\overline{w}}_t{\Vert }_{L^p}\le C\Vert A-\nabla {\overline{v}}_t\otimes \nabla {\overline{v}}_t{\Vert }_{L^p},$$

$$\nabla {\overline{\Phi }}_t=R_t+t^2\nabla {\overline{w}}_t.$$

$${\tilde{u}}_t={\tilde{R}}_t^T{\overline{u}}_t$$

$$\nabla {\tilde{u}}_t=\left(\begin{array}{c}I+t^2\nabla {\tilde{w}}_t\\ t\nabla {\overline{v}}_t\end{array}\right)$$

$$\Vert \nabla {\tilde{w}}_t{\Vert }_{L^p}\le C\Vert A-\nabla {\overline{v}}_t\otimes \nabla {\overline{v}}_t{\Vert }_{L^p}.$$

Full text

Keywords:

$h$-principle, isometric embeddings, Monge-Ampère, von Kármán, nonlinear elasticity

Mathematics Subject Classification:

35J96, 74G20

1 Introduction and main result

The classical $h$-principle of Nash and Kuiper shows that there exist surprisingly many $C^{1}$ solutions $u:\Omega\subset\mathbb{R}^{2}\to\mathbb{R}^{3}$ to the isometric immersion system

[TABLE]

In contrast, classical rigidity results show that, among more regular immersions, being a solution of system (1) is as restrictive a condition as one might expect. A natural question is whether such results extend to classes of $C^{1,\alpha}$ Hölder spaces: for the $h$-principle one seeks the largest possible Hölder exponent $\alpha\in(0,1)$ and for the rigidity the smallest possible one. We refer to [1] and the references therein.

Such a dichotomy between an $h$-principle on one hand and rigidity on the other hand also applies to other PDE systems. A system for which this is to be expected is the system

[TABLE]

for $w:\Omega\to\mathbb{R}^{2}$ and $v:\Omega\to\mathbb{R}$. This system arises naturally as a constraint in von Kármán theories (cf. [3]) in certain energy regimes. In that context, $w$ describes the in-plane displacement and $v$ the out-of-plane displacement. It is clearly related to the Monge-Ampère equation $\det\nabla^{2}v=\operatorname{curl}\operatorname{curl}A$. We refer e.g. to [3] for some details on this.

System (2) is closely related to (1), and it was shown in [4] that the convex integration construction in [1] can indeed be adapted to obtain the same statement for system (2). In this paper we show how the close connection between (2) and (1) can be used to derive $C^{1,\alpha}$ $h$-principles for (2) directly from similar results for (1), without having to repeat the construction.

From now on $\Omega\subset\mathbb{R}^{2}$ denotes a bounded and simply connected domain with a smooth boundary. Our main result is the following:

Theorem 1.1**.**

Let $p\geq 2$, let $\beta\in(0,1)$ and let

[TABLE]

*Then there exists $C>0$ such that the following is true:

For any $v\in C^{2}(\overline{\Omega})$, $w\in C^{2}(\overline{\Omega},\mathbb{R}^{2})$, $A\in C^{0,\beta}(\overline{\Omega},\mathbb{R}^{2\times 2}_{\operatorname{sym}})$ there exist $\overline{v}\in C^{1,\alpha}(\overline{\Omega})$ and $\overline{w}\in C^{1,\alpha}(\overline{\Omega},\mathbb{R}^{2})$ with*

[TABLE]

and

[TABLE]

and such that

[TABLE]

Remarks.

Theorem 1.1 is a variant of [1, Theorem 1]. It allows to improve a $C^{1}$ $h$-principle to a $C^{1,\alpha}$ $h$-principle, cf. Corollary 3.2. 2. 2.

A variant of Theorem 1.1 was stated in [4]. Theorem 1.1 is more general in that does not require $2\operatorname{sym}\nabla w+\nabla v\otimes\nabla v$ to be close to $A$. On the other hand it only yields $L^{p}$ rather than uniform bounds on $\nabla w$. (For the actual convex integration result, however, this is immaterial. See Corollary 3.2 below.) The main difference to [4] is that our short proof derives Theorem 1.1 directly from the corresponding result for isometric immersions [1], therefore avoiding the need of adapting each step of the construction in [1].

Notation.

For $n\in\mathbb{N}$, we denote by $\mathbb{R}^{n\times n}_{\operatorname{sym}}$ the set of symmetric $n\times n$ matrices. By $e$ we denote the standard Riemannian metric on $\mathbb{R}^{n}$. Given an immersion $u$ into $\mathbb{R}^{n}$, we denote by $u^{*}e$ the pullback-metric, so that in coordinates

[TABLE]

For $k=0,1,...$ we denote the usual $C^{k}$ norm by $\|u\|_{k}$. For $\beta\in(0,1)$ the Hölder seminorm $[u]_{\beta}$ is defined to be the infimum over all $C$ such that

[TABLE]

The unit matrix is denoted by $I$.

2 $h$-principle for isometric immersions

An inspection of the proof in [1] shows that in that paper the following more detailed version of [1, Theorem 1] is proven:

Proposition 2.1**.**

Let $n\in\mathbb{N}$, $\beta\in(0,1)$,

[TABLE]

*let $g_{0}\in\mathbb{R}^{n\times n}_{\operatorname{sym}}$ be positive definite and let $U\subset\mathbb{R}^{n}$ a smoothly bounded domain. There exist $\varepsilon_{0}$, $C$, $r>0$ such that for all $\theta$, $\mu$, $\delta\in(0,\infty)$ satisfying $\mu\geq\delta$ and $\delta^{\beta-2}\mu^{-\beta}\theta^{2}\leq\varepsilon_{0}$ the following holds:

If $g\in C^{0,\beta}(\overline{U},\mathbb{R}^{n\times n}_{\operatorname{sym}})$ satisfies*

[TABLE]

and if $u\in C^{2}(\overline{U},\mathbb{R}^{n+1})$ satisfies

[TABLE]

then there exists an isometric immersion $\overline{u}\in C^{1,\alpha}(\overline{U},\mathbb{R}^{n+1})$ of $g$ with

[TABLE]

3 $h$-principle for von Kármán constraints

Proposition 3.1**.**

Theorem 1.1 is true provided that, in addition, $w=0$.

Theorem 1.1 follows at once from Proposition 3.1. For the readers’ convenience we include the details:

Proof of Theorem 1.1.

Applying Proposition 3.1 with $\widetilde{A}=A-2\operatorname{sym}\nabla w$, we obtain $\overline{v}\in C^{1,\alpha}(\overline{\Omega})$ and $\widetilde{w}\in C^{1,\alpha}(\overline{\Omega},\mathbb{R}^{2})$ satisfying

[TABLE]

and

[TABLE]

and $2\operatorname{sym}\nabla\widetilde{w}+\nabla\overline{v}\otimes\nabla\overline{v}=A-2\operatorname{sym}\nabla w.$ Hence by the definition of $\widetilde{A}$ the claim follows with $\overline{w}=w+\widetilde{w}$. ∎

Proof of Proposition 3.1.

Set $g_{0}=I$. For every $t>0$ define $g_{t}=I+t^{2}A$. So $[g_{t}]_{\beta}=t^{2}[A]_{\beta}$. Setting $\theta_{t}=\sqrt{1+[A]_{\beta}}\cdot t$, we see that estimate (7) (with index $t$; we omit this remark in what follows) is satisfied. And (6) is satisfied for any $r>0$, provided that $t<t_{0}$, where $t_{0}\in(0,\infty]$ is defined by

[TABLE]

Define $u_{t}:\Omega\to\mathbb{R}^{3}$ by

[TABLE]

and define $D\geq 0$ by $\|\nabla v\otimes\nabla v-A\|_{0}=D^{2}/2$. We may assume that $D>0$, because if $D=0$ then there is nothing to prove. Define $\delta_{t}=Dt$.

We have

[TABLE]

Hence (8) is satisfied.

Finally, for $M\geq 2(1+\|\nabla^{2}v\|_{0}+D)$ and setting $\mu_{t}=Mt$, estimate (9) and $\mu_{t}\geq\delta_{t}$ are satisfied. On the other hand,

[TABLE]

If $M^{\beta}$ exceeds $\varepsilon_{0}^{-1}(1+[A]_{\beta})D^{\beta-2}$, then the right-hand side of (10) does not exceed $\varepsilon_{0}$; here $\varepsilon_{0}$ is the constant from Proposition 2.1.

Hence for every $t\in(0,t_{0})$ Proposition 2.1 furnishes isometric immersions $\overline{u}_{t}\in C^{1,\alpha}(\overline{\Omega},\mathbb{R}^{3})$ of $g_{t}$ satisfying

[TABLE]

Define $\overline{\Phi}_{t}:\Omega\to\mathbb{R}^{2}$ and $\overline{v}_{t}:\Omega\to\mathbb{R}$ by

[TABLE]

Then (11) imply that

[TABLE]

for all $t\in(0,t_{0})$. And $\|\nabla\overline{\Phi}_{t}-I\|_{0}\leq CDt$. In particular, $\det\nabla\overline{\Phi}_{t}>0$ for $t>0$ small enough. Moreover, since $\overline{u}_{t}^{*}e=I+t^{2}A$,

[TABLE]

Hence

[TABLE]

Since $\det\nabla\overline{\Phi}_{t}>0$, we have almost everywhere

[TABLE]

Hence by FJM-rigidity (cf. [2, Theorem 3.1] and the sentence following its statement) there exists a constant $C$ depending only on $p$ (and on $\Omega$) and there exist $R_{t}\in SO(2)$ as well as $\overline{w}_{t}\in W^{1,p}(\Omega,\mathbb{R}^{2})$ such that, for $t>0$ small enough,

[TABLE]

and such that

[TABLE]

Denoting by $\widetilde{R}_{t}\in SO(3)$ the matrix with rotation axis $(0,0,1)^{T}$ and in-plane rotation $R_{t}$, define

[TABLE]

and $\widetilde{w}_{t}=R_{t}^{T}\overline{w}_{t}$. Then

[TABLE]

and

[TABLE]

Since $\widetilde{u}_{t}$ is an isometric immersion of $g_{t}$, we have

[TABLE]

By (12) and (14) there exists a sequence $t\to 0$ and $\overline{v}\in C^{1,\alpha}$ such that $\nabla\overline{v}_{t}\to\nabla\overline{v}$ uniformly and such that $\nabla\widetilde{w}_{t}$ converges weakly in $L^{p}$ to the gradient of some $\overline{w}\in W^{1,p}$. By (14), the matrix fields $(\nabla\widetilde{w}_{t})^{T}(\nabla\widetilde{w}_{t})$ remain uniformly bounded in $L^{1}$ as $t\to 0$. Hence letting $t\to 0$ in (15), we conclude that

[TABLE]

Moreover, taking the limes inferior in (14), we have

[TABLE]

And by (12)

[TABLE]

∎

Combining Theorem 1.1 with a Nash-Kuiper result one obtains the following; see [4] for a similar result.

Corollary 3.2**.**

Let $\beta$, $\alpha$ and $A$ be as in Theorem 1.1. Let $v\in C^{1}(\overline{\Omega})$ and $w\in C^{1}(\overline{\Omega},\mathbb{R}^{2})$ be such that

[TABLE]

for some constant $c>0$, and let $\varepsilon>0$. Then there exist $\overline{v}\in C^{1,\alpha}(\overline{\Omega})$ and $\overline{w}\in C^{1,\alpha}(\overline{\Omega},\mathbb{R}^{2})$ with

[TABLE]

such that

[TABLE]

Proof.

Let $\delta\in(0,1)$ and let $p\in(2,\infty)$. Following [1], let $\widetilde{v}$ respectively $\widetilde{w}\in C^{\infty}(\overline{\Omega})$ be $C^{1}$-close to $v$ respectively $w$. Applying [4, Theorem 2.1] with $\widetilde{v}$, $\widetilde{w}$ and some smooth uniform approximation of $A$, one obtains $\widehat{v}$, $\widehat{w}\in C^{1}$ such that

[TABLE]

By approximation, we may assume that $\widehat{v}$, $\widehat{w}\in C^{2}(\overline{\Omega})$. Applying Theorem 1.1 we obtain $\overline{v}$, $\overline{w}\in C^{1,\alpha}(\overline{\Omega})$ satisfying

[TABLE]

and

[TABLE]

and $2\operatorname{sym}\nabla\overline{w}+\nabla\overline{v}\otimes\nabla\overline{v}=A$. Notice that $\widehat{v}$ can be chosen such that $\|\nabla\widehat{v}\|_{0}\leq C(1+\|\nabla\widetilde{v}\|_{0})\leq C(1+\|\nabla v\|_{0})$ for some constant $C$ independent of $\delta$, cf. [4, Remark 3.3].

The claim now follows from the continuous embedding of $W^{1,p}$ into $C^{0}$, and from the arbitrariness of $\delta$. ∎

Acknowledgements. Both authors acknowledge support by the DFG.