# W 2,p -estimates for surfaces in terms of their two fundamental forms

Philippe G. Ciarlet, Cristinel Mardare

► To cite this version:

Philippe G. Ciarlet, Cristinel Mardare. W 2,p -estimates for surfaces in terms of their two fundamental forms. Comptes Rendus. Mathématique, 2018, 356 (1), pp.85-91. 10.1016/j.crma.2017.12.003 . hal-01727386

HAL Id: hal-01727386

https://hal.sorbonne-universite.fr/hal-01727386

Submitted on 9 Mar 2018

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
The three-dimensional Euclidean space is denoted $\mathbb{E}^3$. The inner product, exterior product, and norm, in $\mathbb{E}^3$ are respec-
tively denoted ·, ∧, and |·| . The set of all proper isometries of $\mathbb{E}^3$ is denoted and defined by

$$
\mathbf{Isom}_+(\mathbb{E}^3) := \{ \mathbf{r} : \mathbb{E}^3 \to \mathbb{E}^3, \mathbf{r}(x) = \mathbf{R}x + \mathbf{a}, x \in \mathbb{E}^3; \mathbf{R} \in \mathbb{O}_+^3, \mathbf{a} \in \mathbb{E}^3 \},
$$

where $\mathbb{O}_+^3$ denotes the set of all real 3 × 3 proper orthogonal matrices.

**Remark 1.** The set $\mathbf{Isom}_+(\mathbb{E}^3)$ is in effect a smooth submanifold of dimension six of the space of all $3 \times 3$ real matrices and its tangent space at the identity mapping $\mathbf{id} \in \mathbf{Isom}_+(\mathbb{E}^3)$ is the space of all “infinitesimal rigid displacements of $\mathbb{E}^3$”, which is denoted and defined by

$$
\mathbf{Rig}(\mathbb{E}^3) = \mathcal{T}_{\mathbf{id}}\mathbf{Isom}_+(\mathbb{E}^3) := \{ \boldsymbol{\zeta} : \mathbb{E}^3 \to \mathbb{E}^3, \boldsymbol{\zeta}(x) = \mathbf{A}x + \mathbf{b}, x \in \mathbb{E}^3; \mathbf{A} \in \mathbb{A}^3, \mathbf{b} \in \mathbb{E}^3 \},
$$

where $\mathbb{A}^3$ denotes the set of all real 3 × 3 antisymmetric matrices. $\square$

Given an open subset $\omega$ of $\mathbb{R}^2$, we let $y = (y_\alpha)$ denote a generic point in $\omega$, and we let $\partial_\alpha := \partial/\partial y_\alpha$ and $\partial_{\alpha\beta} := \partial^2/\partial y_\alpha \partial y_\beta$.

The space of distributions over an open subset $\omega$ of $\mathbb{R}^2$ is denoted $\mathcal{D}'(\omega)$. For each integer $m \ge 1$ and each real number $p \ge 1$, $C^m(\omega)$ denotes the subspace of $C^0(\omega)$ of functions that possess continuous partial derivatives up to order $m$, and $W^{m,p}(\omega)$ denotes the usual Sobolev space.

The notation $L_{\text{loc}}^{p}(\omega)$, resp. $W_{\text{loc}}^{m,p}(\omega)$, denotes the space of functions $f: \omega \to \mathbb{R}$ such that $f|_{U} \in L^{p}(U)$, resp. $f|_{U} \in W^{m,p}(U)$, for all open sets $U \in \omega$, where $f|_{U}$ denotes the restriction of $f$ to $U$ and the notation $U \in \omega$ means that the closure of the set $U$ is a compact subset of $\omega$. Given any finite dimensional real space $\mathbb{Y}$, the notation $L_{\text{loc}}^{p}(\omega; \mathbb{Y})$, resp. $W_{\text{loc}}^{1,p}(\omega; \mathbb{Y})$, denotes the space of $\mathbb{Y}$-valued fields with components in $L_{\text{loc}}^{p}(\omega)$, resp. $W_{\text{loc}}^{1,p}(\omega)$. Other similar notations with self-explanatory definitions will be used.

An immersion from $\omega$ into $\mathbb{E}^3$ is a smooth enough mapping $\boldsymbol{\theta} : \omega \to \mathbb{E}^3$ such that the two vector fields $\partial_{\alpha} \boldsymbol{\theta} : \omega \to \mathbb{E}^3$ are linearly independent at each point of $\omega$. Given an immersion $\boldsymbol{\theta} : \omega \to \mathbb{E}^3$, define the functions

$$
\hat{a}_{\alpha\beta}(\boldsymbol{\theta}) := \hat{a}_{\alpha}(\boldsymbol{\theta}) \cdot \hat{a}_{\beta}(\boldsymbol{\theta}) \quad \text{and} \quad \hat{b}_{\alpha\beta}(\boldsymbol{\theta}) := \partial_{\alpha} \hat{a}_{\beta}(\boldsymbol{\theta}) \cdot \hat{a}_{3}(\boldsymbol{\theta}),
$$

where

$$
\hat{a}_{\alpha}(\boldsymbol{\theta}) := \partial_{\alpha} \boldsymbol{\theta} \quad \text{and} \quad \hat{a}_{3}(\boldsymbol{\theta}) := \frac{\partial_{1} \boldsymbol{\theta} \wedge \partial_{2} \boldsymbol{\theta}}{|\partial_{1} \boldsymbol{\theta} \wedge \partial_{2} \boldsymbol{\theta}|}.
$$

The image $S = \boldsymbol{\theta}(\omega)$ is thus a surface in $\mathbb{E}^3$ and the functions $\hat{a}_{\alpha\beta}(\boldsymbol{\theta})$ and $\hat{b}_{\alpha\beta}(\boldsymbol{\theta})$ are the covariant components of the first and second fundamental forms of $S$.

The space of real 2 × 2 symmetric matrices is denoted $\mathbb{S}^2$; its subset formed by all positive-definite matrices is denoted $\mathbb{S}_+^2$.

An open subset $\omega$ of $\mathbb{R}^2$ satisfies the *uniform interior cone property* if there exists a bounded open cone $V \subset \mathbb{R}^2$ such that any point $y \in \omega$ is the vertex of a cone $V_y$ congruent with $V$ and contained in $\omega$. An open subset $\omega$ of $\mathbb{R}^2$ is a *domain* if it is bounded and has a Lipschitz-continuous boundary.

Detailed proofs of the results announced here will be found in [4].

2. The fundamental theorem of surface theory in the spaces $W_{\text{loc}}^{2,p}(\omega)$ and $W^{2,p}(\omega)$

The *fundamental theorem of surface theory*, which is classically established in the spaces of continuously differentiable functions (cf., e.g., [5, Theorem 3.8.8], [1, Appendix to Chapter 4], [2, Theorems 8.16-1 and 8.17-1]), has been shown to hold in function spaces with little regularity, according to the following remarkable result, due to S. Mardare [6, Theorem 9]:

**Theorem 1.** Let $\omega$ be a simply-connected open subset of $\mathbb{R}^2$, let $p > 2$, and let a matrix field $(a_{\alpha\beta}) \in W_{\text{loc}}^{1,p}(\omega; \mathbb{S}_+^2)$ and a matrix field $(b_{\alpha\beta}) \in L_{\text{loc}}^p(\omega; \mathbb{S}^2)$ be given that satisfy the Gauss and Codazzi–Mainardi equations, viz.

$$
R_{\alpha\beta\tau}^{\sigma} := \partial_{\tau} \Gamma_{\alpha\beta}^{\sigma} - \partial_{\beta} \Gamma_{\alpha\tau}^{\sigma} + \Gamma_{\alpha\beta}^{\gamma} \Gamma_{\tau\gamma}^{\sigma} - \Gamma_{\alpha\tau}^{\gamma} \Gamma_{\beta\gamma}^{\sigma} - b_{\alpha\beta} b_{\tau}^{\sigma} + b_{\alpha\tau} b_{\beta}^{\sigma} = 0 \quad \text{in } \mathcal{D}'(\omega)
$$

and

$$
R_{\alpha\beta\tau}^{3} := \partial_{\tau} b_{\alpha\beta} - \partial_{\beta} b_{\alpha\tau} + \Gamma_{\alpha\beta}^{\gamma} b_{\tau\gamma} - \Gamma_{\alpha\tau}^{\gamma} b_{\beta\gamma} = 0 \text{ in } \mathcal{D}'(\omega),
$$

where the functions $\Gamma_{\alpha\beta}^{\sigma} \in L_{\text{loc}}^{p}(\omega)$ and $b_{\alpha}^{\sigma} \in L_{\text{loc}}^{p}(\omega)$ are defined by

$$
\Gamma_{\alpha\beta}^{\sigma} := \frac{1}{2} a^{\sigma\tau} (\partial_{\alpha} a_{\beta\tau} + \partial_{\beta} a_{\alpha\tau} - \partial_{\tau} a_{\alpha\beta}) \text{ and } b_{\beta}^{\sigma} := a^{\sigma\tau} b_{\tau\beta}, \text{ where } (a^{\sigma\tau}) := (a_{\alpha\beta})^{-1}.
$$
Then there exists an immersion $\theta \in W_{\text{loc}}^{2,p}(\omega; \mathbb{E}^3)$ such that

$$ \hat{a}_{\alpha\beta}(\theta) = a_{\alpha\beta} \text{ and } \hat{b}_{\alpha\beta}(\theta) = b_{\alpha\beta} \text{ a.e. in } \omega. $$

Besides, an immersion $\psi \in W_{\text{loc}}^{2,p}(\omega; \mathbb{E}^3)$ satisfies

$$ \hat{a}_{\alpha\beta}(\psi) = \hat{a}_{\alpha\beta}(\theta) \text{ and } \hat{b}_{\alpha\beta}(\psi) = \hat{b}_{\alpha\beta}(\theta) \text{ a.e. in } \omega $$

if and only if there exists an isometry $\mathbf{r} \in \mathbf{Isom}_+(\mathbb{E}^3)$ such that

$$ \psi = \mathbf{r} \circ \theta \text{ in } \omega. \quad \square $$

Our first objective (Theorem 2) consists in showing that an existence and uniqueness theorem similar to Theorem 1 holds in the spaces $W^{m,p}(\omega)$ instead of the spaces $W_{\text{loc}}^{m,p}(\omega)$ if the open set $\omega$ is in addition a domain.

**Theorem 2.** Let $\omega$ be a simply-connected domain in $\mathbb{R}^2$, let $p > 2$, and let a matrix field $(a_{\alpha\beta}) \in W^{1,p}(\omega; \mathbb{S}^2)$ and a matrix field $(b_{\alpha\beta}) \in L^p(\omega; \mathbb{S}^2)$ be given that satisfy the equations

$$ R_{\alpha\beta\tau}^{\sigma} = 0 \text{ and } R_{\alpha\beta\tau}^{3} = 0 \text{ in } \mathcal{D}'(\omega). $$

Then there exists an immersion $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$ such that

$$ \hat{a}_{\alpha\beta}(\theta) = a_{\alpha\beta} \text{ and } \hat{b}_{\alpha\beta}(\theta) = b_{\alpha\beta} \text{ a.e. in } \omega. $$

Besides, an immersion $\psi \in W^{2,p}(\omega; \mathbb{E}^3)$ satisfies

$$ \hat{a}_{\alpha\beta}(\psi) = \hat{a}_{\alpha\beta}(\theta) \text{ and } \hat{b}_{\alpha\beta}(\psi) = \hat{b}_{\alpha\beta}(\theta) \text{ a.e. in } \omega $$

if and only if there exists an isometry $\mathbf{r} \in \mathbf{Isom}_+(\mathbb{E}^3)$ such that

$$ \psi = \mathbf{r} \circ \theta \text{ in } \omega. \quad \square $$

**Sketch of proof.** Since $p > 2$ and $\omega$ is a domain, $W^{1,p}(\omega)$ is a Banach algebra and the canonical injection from $W^{1,p}(\omega)$ into $C^0(\bar{\omega})$ is continuous. Combining these two observations with the Gauss equations

$$ \partial_{\alpha} \hat{a}_{\beta}(\theta) = \Gamma_{\alpha\beta}^{\sigma} \hat{a}_{\sigma}(\theta) + b_{\alpha\beta} \hat{a}_{3}(\theta) \text{ a.e. in } \omega $$

and the relations

$$ |\hat{a}_{\alpha}(\theta)| = \sqrt{a_{\alpha\alpha}} \text{ (no summation on } \alpha \text{ here) and } |\hat{a}_{3}(\theta)| = 1 \text{ a.e. in } \omega, $$

where $\theta \in W_{\text{loc}}^{2,p}(\omega; \mathbb{E}^3)$ denotes the immersion found in Theorem 1 and the functions $\Gamma_{\alpha\beta}^{\sigma}$ are defined as in Theorem 1 (in effect the Christoffel symbols associated with $\theta$), shows that the three vector fields $\hat{a}_i(\theta)$ belong to $L^{\infty}(\omega; \mathbb{E}^3)$, which in turn implies that $\partial_{\alpha}\theta \in L^{\infty}(\omega; \mathbb{E}^3)$ and $\partial_{\alpha\beta}\theta \in L^p(\omega; \mathbb{E}^3)$. It is then an easy matter to conclude that $\theta \in L^p(\omega; \mathbb{E}^3)$, hence that $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$. The uniqueness up to isometries follows immediately from Theorem 1. $\square$

### 3. A nonlinear Korn inequality for surfaces of class $W^{2,p}$

The second objective of this Note is to complement the existence and uniqueness result of Theorem 2 by a stability result (Theorem 3 below), showing that the distance modulo a proper isometry between two surfaces in $W^{2,p}$-norm is bounded by the distance between their first fundamental forms in the $W^{1,p}$-norm and the distance between their second fundamental forms in the $L^p$-norm. A notation such as $c = c(\omega, p, \varepsilon)$ means that $c$ is a real constant that depends on $\omega, p$ and $\varepsilon$.

**Theorem 3.** Let $\omega$ be a bounded and connected open subset of $\mathbb{R}^2$ that satisfies the uniform interior cone property. Given any $p > 2$ and $\varepsilon > 0$, let

$$ V_{\varepsilon}(\omega; \mathbb{E}^3) := \{ \theta \in W^{2,p}(\omega; \mathbb{E}^3); \|\theta\|_{W^{2,p}(\omega; \mathbb{E}^3)} \le 1/\varepsilon \text{ and } |\partial_1 \theta \wedge \partial_2 \theta| \ge \varepsilon \text{ in } \omega \}. $$

Then there exists a constant $c = c(\omega, p, \varepsilon)$ such that

$$ \inf_{\mathbf{r} \in \mathbf{Isom}_+(\mathbb{E}^3)} \|\varphi - \mathbf{r} \circ \psi\|_{W^{2,p}(\omega; \mathbb{E}^3)} \le c \left\{ \| (\hat{a}_{\alpha\beta}(\varphi) - \hat{a}_{\alpha\beta}(\psi))\|_{W^{1,p}(\omega; \mathbb{S}^2)} + \| (\hat{b}_{\alpha\beta}(\varphi) - \hat{b}_{\alpha\beta}(\psi))\|_{L^p(\omega; \mathbb{S}^2)} \right\} $$

for all $\varphi \in V_{\varepsilon}(\omega; \mathbb{E}^3)$ and $\psi \in V_{\varepsilon}(\omega; \mathbb{E}^3)$. $\square$
**Remark 2.** The above inequality can indeed be seen as a nonlinear Korn inequality for surfaces of class $W^{2,p}$, since a formal linearization (such a linearization consists first in letting in the above nonlinear inequality $\varphi := \theta + \eta$ and $\psi := \theta$, where $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$ is a given immersion considered as “fixed”, and $\eta \in W^{2,p}(\omega; \mathbb{E}^3)$ is an arbitrary vector field, then in canceling all the terms that depend nonlinearly on $\eta$) yields the following linear Korn inequality on the surface $S = \theta(\omega)$: There exists a constant $c_0 = c_0(\theta, \omega)$ such that (the space **Rig**(E³) is defined in **Remark 1**)

$$ \inf_{\zeta \in \text{Rig}(\mathbb{E}^3)} \|\eta - \zeta\|_{W^{2,p}(\omega; \mathbb{E}^3)} \le c_0 \left\{ \|(\gamma_{\alpha\beta}(\eta))\|_{W^{1,p}(\omega; \mathbb{S}^2)} + \|(\rho_{\alpha\beta}(\eta))\|_{L^p(\omega; \mathbb{S}^2)} \right\} \quad \text{for all } \eta \in W^{2,p}(\omega; \mathbb{E}^3), $$

where

$$ \gamma_{\alpha\beta}(\eta) := \frac{1}{2} [\hat{a}_{\alpha\beta}(\theta + \eta) - \hat{a}_{\alpha\beta}(\theta)]^{\text{lin}} \quad \text{and} \quad \rho_{\alpha\beta}(\eta) := [\hat{b}_{\alpha\beta}(\theta + \eta) - \hat{b}_{\alpha\beta}(\theta)]^{\text{lin}} $$

designate the linear parts with respect to $\eta$ of the tensors appearing in the right-hand side of the inequality of **Theorem 3**. $\square$

The proof of **Theorem 3** relies on a comparison theorem between solutions to general Pfaff systems due to the first author and S. Mardare (see **Theorem 3.1** and **Remark 3.1** in [3] and **Theorem 4.1** in [7]), which we state below only in the particular case needed here. The notations $\mathbb{M}^3$ and $| \cdot |$ used in the next theorem respectively denote the space of 3 × 3 real matrices and the Frobenius norm in this space. The notation $(\mathbf{a} | \mathbf{b} | \mathbf{c})$ denotes the matrix in $\mathbb{M}^3$ with column vectors $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{E}^3$.

**Theorem 4.** Let $\omega$ be a bounded and connected open subset of $\mathbb{R}^2$ that satisfies the uniform interior cone property. Given any $p > 2$, $\varepsilon > 0$, and $y_0 \in \omega$, there exists a constant $c_1 = c_1(\omega, p, \varepsilon, y_0)$ such that

$$ \|\mathbf{F} - \tilde{\mathbf{F}}\|_{W^{1,p}(\omega; \mathbb{M}^3)} \le c_1 (|\mathbf{F}(y_0) - \tilde{\mathbf{F}}(y_0)| + \sum_{\alpha} \|\mathbf{\Gamma}_{\alpha} - \tilde{\mathbf{\Gamma}}_{\alpha}\|_{L^p(\omega; \mathbb{M}^3)}) $$

for all matrix fields $\mathbf{F}, \tilde{\mathbf{F}} \in W^{1,p}(\omega; \mathbb{M}^3)$ and $\mathbf{\Gamma}_{\alpha}, \tilde{\mathbf{\Gamma}}_{\alpha} \in L^p(\omega; \mathbb{M}^3)$ that satisfy

$$ |\mathbf{F}(y_0)| + \sum_{\alpha} \|\mathbf{\Gamma}_{\alpha}\|_{L^p(\omega; \mathbb{M}^3)} \le \frac{1}{\varepsilon} \quad \text{and} \quad |\tilde{\mathbf{F}}(y_0)| + \sum_{\alpha} \|\tilde{\mathbf{\Gamma}}_{\alpha}\|_{L^p(\omega; \mathbb{M}^3)} \le \frac{1}{\varepsilon}, $$

and

$$ \partial_{\alpha} \mathbf{F} = \mathbf{F} \mathbf{\Gamma}_{\alpha} \quad \text{and} \quad \partial_{\alpha} \tilde{\mathbf{F}} = \tilde{\mathbf{F}} \tilde{\mathbf{\Gamma}}_{\alpha} \quad \text{a.e. in } \omega. \quad \square $$

**Sketch of the proof of Theorem 3.** With any immersion $\varphi \in W^{2,p}(\omega; \mathbb{E}^3)$, we associate: the proper isometry $\mathbf{r}(\varphi, y_0)$ of $\mathbb{E}^3$ defined by

$$ \mathbf{r}(\varphi, y_0)(x) := (\mathbf{B}^T \mathbf{B})^{1/2} \mathbf{B}^{-1} (x - \varphi(y_0)) \quad \text{for all } x \in \mathbb{E}^3, $$

where

$$ \mathbf{B} := (\hat{\mathbf{a}}_1(\varphi)(y_0) | \hat{\mathbf{a}}_2(\varphi)(y_0) | \hat{\mathbf{a}}_3(\varphi)(y_0)); $$

the immersion

$$ \theta(\varphi, y_0) := \mathbf{r}(\varphi, y_0) \circ \varphi \in W^{2,p}(\omega; \mathbb{E}^3); $$

and the matrix fields

$$ \mathbf{F}(\varphi, y_0) := (\hat{\mathbf{a}}_1(\theta(\varphi, y_0)) | \hat{\mathbf{a}}_2(\theta(\varphi, y_0)) | \hat{\mathbf{a}}_3(\theta(\varphi, y_0))) $$

and

$$ \mathbf{A}(\varphi) := \begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \text{and} \quad \mathbf{\Gamma}_{\alpha}(\varphi) := \begin{pmatrix} \Gamma_{\alpha 1}^{1} & \Gamma_{\alpha 2}^{1} & -b_{\alpha}^{1} \\ \Gamma_{\alpha 1}^{2} & \Gamma_{\alpha 2}^{2} & -b_{\alpha}^{2} \\ b_{\alpha 1} & b_{\alpha 2} & 0 \end{pmatrix}, $$

where

$$ a_{\alpha\beta} := \hat{a}_{\alpha\beta}(\varphi), \quad b_{\alpha\beta} := \hat{b}_{\alpha\beta}(\varphi), \quad b_{\beta}^{\alpha} := a^{\alpha\sigma} b_{\sigma\beta}, \quad (a^{\sigma\tau}) := (a_{\alpha\beta})^{-1}, $$

and

$$ \Gamma_{\alpha\beta}^{\sigma} := \frac{1}{2} a^{\sigma\tau} (\partial_{\alpha} a_{\beta\tau} + \partial_{\beta} a_{\alpha\tau} - \partial_{\tau} a_{\alpha\beta}). $$
These matrix fields satisfy the Pfaff system

$$ \partial_{\alpha} \mathbf{F}(\varphi, y_0) = \mathbf{F}(\varphi, y_0) \mathbf{\Gamma}_{\alpha}(\varphi) \text{ a.e. in } \omega, $$

and the “initial condition”

$$ (\mathbf{F}(\varphi, y_0))(y_0) = (\mathbf{A}(\varphi)(y_0))^{1/2} \in \mathbb{S}_{>}^3. $$

Note in passing that the above Pfaff system is equivalent to the equations of Gauss and Weingarten associated with the immersion $\theta(\varphi, y_0)$.

In addition, if $\varphi \in V_{\varepsilon}(\omega; \mathbb{E}^3)$ for some $\varepsilon > 0$ (the set $V_{\varepsilon}(\omega; \mathbb{E}^3)$ is defined in the statement of Theorem 3), then

$$ \mathbf{F}(\varphi, y_0) \in W^{1,p}(\omega; \mathbb{S}^3) \text{ and } \mathbf{\Gamma}_{\alpha}(\varphi) \in L^p(\omega; \mathbb{M}^3), $$

and there exists a constant $c_1 = c_1(\omega, p, \varepsilon)$ such that

$$ |(\mathbf{F}(\varphi, y_0))(y_0)| + \|\mathbf{\Gamma}_{\alpha}(\varphi)\|_{L^p(\omega; \mathbb{M}^3)} \le c_1. $$

This allows us to apply Theorem 4 and to deduce that there exists a constant $c_2 = c_2(\omega, y_0, p, \varepsilon)$ such that

$$ \| \mathbf{F}(\varphi, y_0) - \mathbf{F}(\psi, y_0) \|_{W^{1,p}(\omega; \mathbb{M}^3)} \le c_2 \left( |(\mathbf{A}(\varphi))(y_0) - (\mathbf{A}(\psi))(y_0)| + \sum_{\alpha} \| \mathbf{\Gamma}_{\alpha}(\varphi) - \mathbf{\Gamma}_{\alpha}(\psi) \|_{L^p(\omega; \mathbb{M}^3)} \right) $$

for all immersions $\varphi$ and $\psi$ that belong to the set $V_{\varepsilon}(\omega; \mathbb{E}^3)$.

Next, using the expressions of the matrix fields appearing in the right-hand side of the above inequality in terms of the fundamental forms associated with the immersions $\varphi$ and $\psi$, we deduce after a series of straightforward, but somewhat technical, computations that there exist two constants $c_3 = c_3(\omega, p, \varepsilon)$ and $c_4 = c_4(\omega, p, \varepsilon)$ such that

$$ |(\mathbf{A}(\varphi))(y_0) - (\mathbf{A}(\psi))(y_0)| \le c_3 \|(\hat{\alpha}_{\alpha\beta}(\varphi) - \hat{\alpha}_{\alpha\beta}(\psi))\|_{W^{1,p}(\omega; \mathbb{S}^2)}, $$

and

$$ \| \mathbf{\Gamma}_{\alpha}(\varphi) - \mathbf{\Gamma}_{\alpha}(\psi) \|_{L^p(\omega; \mathbb{M}^3)} \le c_4 \left( \| (\hat{\alpha}_{\alpha\beta}(\varphi) - \hat{\alpha}_{\alpha\beta}(\psi)) \|_{W^{1,p}(\omega; \mathbb{S}^2)} + \| (\hat{\beta}_{\alpha\beta}(\varphi) - \hat{\beta}_{\alpha\beta}(\psi)) \|_{L^p(\omega; \mathbb{S}^2)} \right). $$

Finally, the definition of the immersions $\theta(\varphi, y_0)$ and $\theta(\psi, y_0)$ implies that the vector field

$$ \eta := \theta(\varphi, y_0) - \theta(\psi, y_0) \in W^{2,p}(\omega; \mathbb{E}^3) $$

satisfies the Poincaré system (the notation $[ \cdot ]_\alpha$ denotes the $\alpha$-th column vector of the matrix appearing between the brackets)

$$ \partial_{\alpha} \eta = [\mathbf{F}(\varphi, y_0) - \mathbf{F}(\psi, y_0)]_{\alpha} \text{ in } \omega $$

and the “initial condition”

$$ \eta(y_0) = 0. $$

Using an inequality of Poincaré's type, we infer from the above system and initial condition that there exists a constant $c_5 = c_5(\omega, p)$ such that

$$ \|\eta\|_{W^{2,p}(\omega; \mathbb{E}^3)} \le c_5 \|\mathbf{F}(\varphi, y_0) - \mathbf{F}(\psi, y_0)\|_{W^{1,p}(\omega; \mathbb{M}^3)}. $$

The conclusion follows by combining the above inequalities and by noting that, thanks to the invariance under rotations of the Euclidean and Frobenius norms,

$$ \|\eta\|_{W^{2,p}(\omega; \mathbb{E}^3)} = \|\theta(\varphi, y_0) - \theta(\psi, y_0)\|_{W^{2,p}(\omega; \mathbb{E}^3)} \ge \inf_{\mathbf{r} \in \text{Isom}_+(\mathbb{E}^3)} \|\varphi - \mathbf{r} \circ \psi\|_{W^{2,p}(\omega; \mathbb{E}^3)}. \quad \square $$
### 4. Local Lipschitz-continuity of the mapping defining a surface of class W<sup>2,p</sup>, p > 2, in terms of its fundamental forms

Let $\omega$ be an open subset of $\mathbb{R}^2$. Given two symmetric matrix fields

$\mathbf{A} = (a_{\alpha\beta}) \in W_{\text{loc}}^{1,p}(\omega; \mathbb{S}^2)$ and $\mathbf{B} = (b_{\alpha\beta}) \in L_{\text{loc}}^p(\omega; \mathbb{S}^2)$, $p > 2$,

such that $\mathbf{A}(y) \in \mathbb{S}_>^2$ for all $y \in \bar{\omega}$, define the distributions

$$R_{\alpha\beta\tau}^{\sigma}(\mathbf{A}, \mathbf{B}) := \partial_{\tau}\Gamma_{\alpha\beta}^{\sigma} - \partial_{\beta}\Gamma_{\alpha\tau}^{\sigma} + \Gamma_{\alpha\beta}^{\gamma}\Gamma_{\tau\gamma}^{\sigma} - \Gamma_{\alpha\tau}^{\gamma}\Gamma_{\beta\gamma}^{\sigma} - b_{\alpha\beta}b_{\tau}^{\sigma} + b_{\alpha\tau}b_{\beta}^{\sigma} \in \mathcal{D}'(\omega),$$

$$R_{\alpha\beta\tau}^{3}(\mathbf{A}, \mathbf{B}) := \partial_{\tau}b_{\alpha\beta} - \partial_{\beta}b_{\alpha\tau} + \Gamma_{\alpha\beta}^{\gamma}b_{\tau\gamma} - \Gamma_{\alpha\tau}^{\gamma}b_{\beta\gamma} \in \mathcal{D}'(\omega),$$

where

$$\Gamma_{\alpha\beta}^{\sigma} = \Gamma_{\alpha\beta}^{\sigma}(\mathbf{A}) := \frac{1}{2}a^{\sigma\tau}(\partial_{\alpha}a_{\beta\tau} + \partial_{\beta}a_{\alpha\tau} - \partial_{\tau}a_{\alpha\beta}) \in L_{\text{loc}}^{p}(\omega),$$

$$b_{\beta}^{\sigma} := a^{\sigma\tau}b_{\tau\beta} \in L_{\text{loc}}^{p}(\omega), \text{ and } (a^{\sigma\tau}) := (a_{\alpha\beta})^{-1} \in W_{\text{loc}}^{1,p}(\omega).$$

**Remark 3.** The above regularity assumptions on the fields **A** and **B** are the minimal possible in order that the definitions of the distributions $R_{\alpha\beta\tau}^j(\mathbf{A}, \mathbf{B})$ make sense: combined with the Sobolev embedding $W_{\text{loc}}^{1,p}(\omega) \subset C^0(\omega)$, they ensure that $\det \mathbf{A}$ is a continuous positive function over $\omega$, which in turn implies that $a^{\sigma\tau} \in C^0(\omega)$ and so the products appearing in the definitions of $\Gamma_{\alpha\beta}^{\sigma}$ and $b_{\alpha}^{\sigma}$ belong to $L_{\text{loc}}^p(\omega)$; this allows to define the partial derivatives of $\Gamma_{\alpha\beta}^{\sigma}$ and $b_{\alpha}^{\sigma}$ appearing in the above definition of $R_{\alpha\beta\tau}^j(\mathbf{A}, \mathbf{B})$ as distributions in $\mathcal{D}'(\omega)$. □

The third objective of this Note is to establish, as a consequence of the nonlinear Korn inequality of Theorem 3, the following "existence, uniqueness, and stability theorem" for the reconstruction of a surface from its fundamental forms in the spaces $W^{1,p}(\omega; \mathbb{S}^2)$ and $L^p(\omega; \mathbb{S}^2)$.

In Theorem 5 below, the set $\dot{W}^{2,p}(\omega; \mathbb{E}^3)$ is the quotient set of the space $W^{2,p}(\omega; \mathbb{E}^3)$ by the equivalence relation between isometrically equivalent immersions, and the set $\mathcal{T}(\omega)$ is the subset of the space $W^{1,p}(\omega; \mathbb{S}^2) \times L^p(\omega; \mathbb{S}^2)$ formed by all pairs of a positive-definite symmetric matrix field and a symmetric matrix field that satisfy together the equations of Gauss and Codazzi–Mainardi in the distributional sense. As such, the sets $\dot{W}^{2,p}(\omega; \mathbb{E}^3)$ and $\mathcal{T}(\omega)$ are metric spaces equipped respectively with the distances defined by

$$\mathrm{dist}_{\dot{W}^{2,p}(\omega; \mathbb{E}^3)}(\dot{\theta}, \dot{\psi}) := \inf_{\tilde{\theta} \in \tilde{\theta}, \dot{\psi} \in \dot{\psi}} \|\tilde{\theta} - \tilde{\psi}\|_{W^{2,p}(\omega; \mathbb{E}^3)} = \inf_{\mathbf{r} \in \mathrm{Isom}_+(\mathbb{E}^3)} \|\mathbf{\theta} - \mathbf{r} \circ \mathbf{\psi}\|_{W^{2,p}(\omega; \mathbb{E}^3)}$$

for all $\dot{\theta}$ and $\dot{\psi}$ in $\dot{W}^{2,p}(\omega; \mathbb{E}^3)$, and by

$$\mathrm{dist}_{\mathcal{T}(\omega)}((\mathbf{A}, \mathbf{B}), (\tilde{\mathbf{A}}, \tilde{\mathbf{B}})) := \|\mathbf{A} - \tilde{\mathbf{A}}\|_{W^{1,p}(\omega; \mathbb{S}^2)} + \|\mathbf{B} - \tilde{\mathbf{B}}\|_{L^p(\omega; \mathbb{S}^2)}$$

for all $(\mathbf{A}, \mathbf{B})$ and $(\tilde{\mathbf{A}}, \tilde{\mathbf{B}})$ in $\mathcal{T}(\omega)$.

**Theorem 5.** Let $\omega$ be a domain in $\mathbb{R}^2$. Given any $p > 2$, define the sets

$$\dot{W}^{2,p}(\omega; \mathbb{E}^3) := \{\dot{\theta} = \{\mathbf{r} \circ \theta; \mathbf{r} \in \mathrm{Isom}_+(\mathbb{E}^3)\}; \theta \in W^{2,p}(\omega; \mathbb{E}^3)\}$$

and

$$\mathcal{T}(\omega) := \{(\mathbf{A}, \mathbf{B}) \in W^{1,p}(\omega; \mathbb{S}^2) \times L^p(\omega; \mathbb{S}^2); \mathbf{A}(y) \in \mathbb{S}_>^2 \text{ at each } y \in \bar{\omega}, R_{\alpha\beta\tau}^j(\mathbf{A}, \mathbf{B}) = 0 \text{ in } \mathcal{D}'(\omega)\}.$$

Then the following assertions are true:

(a) Two matrix fields $\mathbf{A} = (a_{\alpha\beta})$ and $\mathbf{B} = (b_{\alpha\beta})$ satisfy

$$(\mathbf{A}, \mathbf{B}) \in \mathcal{T}(\omega)$$

if and only if there exists an immersion $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$ such that

$$\hat{a}_{\alpha\beta}(\theta) = a_{\alpha\beta} \text{ in } \omega \text{ and } \hat{b}_{\alpha\beta}(\theta) = b_{\alpha\beta} \text{ a.e. in } \omega.$$

(b) Two immersions $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$ and $\psi \in W^{2,p}(\omega; \mathbb{E}^3)$ satisfy the relations

$$\hat{a}_{\alpha\beta}(\theta) = \hat{a}_{\alpha\beta}(\psi) \text{ in } \omega \text{ and } \hat{b}_{\alpha\beta}(\theta) = \hat{b}_{\alpha\beta}(\psi) \text{ a.e. in } \omega.$$
if and only if there exists a proper isometry $\mathbf{r}$ of $\mathbb{E}^3$ such that

$$ \psi = \mathbf{r} \circ \theta \text{ in } \omega. $$

(c) The mapping defined by (a) and (b), namely

$$ \mathcal{G}: (\mathbf{A}, \mathbf{B}) \in \mathbb{T}(\omega) \to \mathcal{G}((\mathbf{A}, \mathbf{B})) := \dot{\theta} \in \dot{\mathbb{W}}^{2,p}(\omega; \mathbb{E}^3), $$

where $\theta \in W^{2,p}(\omega; \mathbb{E}^3)$ is any immersion that satisfies

$$ (\hat{a}_{\alpha\beta}(\theta)) = \mathbf{A} \text{ and } (\hat{b}_{\alpha\beta}(\theta)) = \mathbf{B} \text{ a.e. in } \omega, $$

is locally Lipschitz-continuous. □

**Sketch of proof.** Parts (a) and (b) are just a re-statement of Theorem 2. Otherwise, the rest of the proof follows a strategy introduced by the first author and S. Mardare in [3]. More precisely, part (c) of Theorem 5 is deduced from Theorem 3 as follows.

On the one hand, the Sobolev embedding $W^{1,p}(\omega) \subset C^0(\bar{\omega})$ implies that, given any $(\mathbf{A}, \mathbf{B}) \in \mathbb{T}(\omega)$, there exists $\delta = \delta(\mathbf{A}, \mathbf{B}) > 0$ such that the set

$$ \mathbb{T}_{\delta}(\omega) := \{ (\tilde{\mathbf{A}}, \tilde{\mathbf{B}}) \in \mathbb{T}(\omega); \det \tilde{\mathbf{A}} \ge \delta \text{ in } \omega, \|\tilde{\mathbf{A}}\|_{W^{1,p}(\omega;\mathbb{S}^2)} \le 1/\delta, \text{ and } \|\tilde{\mathbf{B}}\|_{L^p(\omega;\mathbb{S}^2)} \le 1/\delta \} $$

is a neighborhood of $(\mathbf{A}, \mathbf{B})$ in the metric space $\mathbb{T}(\omega)$. It also implies that

$$ \mathbb{T}(\omega) = \bigcup_{\delta>0} \mathbb{T}_{\delta}(\omega). $$

Besides, for each $\delta > 0$, there exists $\varepsilon(\delta) > 0$ such that

$$ \mathcal{G}(\mathbb{T}_{\delta}(\omega)) \subset \{ \dot{\theta} \in \dot{\mathbb{W}}^{2,p}(\omega; \mathbb{E}^3); \theta \in V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3) \}, $$

where $\mathcal{G}$ denotes the mapping defined in part (c) of the statement of the theorem and $V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3)$ is defined as in Theorem 3.

On the other hand, Theorem 3 implies that there exists a constant $c = c(\omega, p, \varepsilon(\delta))$ such that

$$ \inf_{\mathbf{r} \in \text{Isom}_+(\mathbb{E}^3)} \|\varphi - \mathbf{r} \circ \psi\|_{W^{2,p}(\omega; \mathbb{E}^3)} \le c \left\{ \| (\hat{a}_{\alpha\beta}(\varphi) - \hat{a}_{\alpha\beta}(\psi))\|_{W^{1,p}(\omega;\mathbb{S}^2)} + \| (\hat{b}_{\alpha\beta}(\varphi) - \hat{b}_{\alpha\beta}(\psi))\|_{L^p(\omega;\mathbb{S}^2)} \right\} $$

for all mappings $\varphi \in V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3)$ and $\psi \in V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3)$ (note that Theorem 3 can be applied under the assumptions of Theorem 5 since a domain satisfies the uniform interior cone property).

We then infer from the observations above that, given any mappings $\varphi \in V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3)$ and $\tilde{\varphi} \in V_{\varepsilon(\delta)}(\omega; \mathbb{E}^3)$ such that $\dot{\varphi} = \mathcal{G}(\mathbf{A}, \mathbf{B})$ and $\dot{\tilde{\varphi}} = \mathcal{G}(\tilde{\mathbf{A}}, \tilde{\mathbf{B}})$ for some $(\mathbf{A}, \mathbf{B}) \in \mathbb{T}_{\delta}(\omega)$ and $(\tilde{\mathbf{A}}, \tilde{\mathbf{B}}) \in \mathbb{T}_{\delta}(\omega)$,

$$ \mathrm{dist}_{\dot{\mathbb{W}}^{2,p}(\omega;\mathbb{E}^3)}(\dot{\varphi}, \tilde{\varphi}) \le c \mathrm{dist}_{\mathbb{T}(\omega)}((\mathbf{A}, \mathbf{B}), (\tilde{\mathbf{A}}, \tilde{\mathbf{B}})). $$

This shows that the restriction of the mapping $\mathcal{G}$ to the set $\mathbb{T}_{\delta}(\omega)$ is Lipschitz-continuous. □

## Acknowledgements

The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. 9042388, CityU 11305716].

## References

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