Geometry, analysis, and morphogenesis: Problems and prospects

Abstract

The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics, and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, which brings together analysis, geometry, and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discuss open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.

1. Introduction

A walk in the garden, a visit to the zoo, or watching a nature documentary reminds us of the remarkable range of living forms on our planet. How these shapes come to be is a question that has interested scientists for eons, and yet it is only over the last century that we have finally begun to grapple with the framework for morphogenesis, a subject that naturally brings together biologists, physicists, and mathematicians. This confluence of approaches is the basis for a book, equally lauded for both its substance and its scientific style, D’Arcy Thompson’s opus, On growth and form 116, where the author says:

An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates vary, proportions change, and the whole configuration alters accordingly.

From both mathematical and mechanical perspectives, this suggests a simple principle: differential growth in a body leads to residual strains that will generically result in changes in the shape of a tissue, organ, or body. Surprisingly then, it is only recently that this principle has been taken up seriously by both experimental and theoretical communities as a viable candidate for patterning at the cellular and tissue level, perhaps because of the dual difficulty of measuring and of calculating the mechanical causes and consequences of these effects. Nevertheless, with an increasing number of testable predictions and high throughput imaging in space-time, this geometric and mechanical perspective on morphogenesis has begun to be viewed as a complement to the biochemical aspects of morphogenesis, as famously exemplified by the work of Alan Turing in his prescient paper, The chemical basis for morphogenesis 119. It is worth pointing out that differential diffusion and growth are only parts of an entire spectrum of mechanisms involved in morphogenesis that include differential adhesion, differential mobility, differential affinity, and differential activity, all of which we must eventually come to grips with to truly understand the development and evolution of biological shape.

In this review, we consider the interplay between geometry, analysis, and morphogenesis of thin surfaces driven by three motivations: the allure of quantifying the aesthetic seen in examples such as flowers, the hope of explaining the origin of shape in biological systems, and the promise of mimicking them in artificial systems 64106. While these issues also arise in three-dimensional tissues in such examples as the folding of the brain 113114 or the looping of the gut 105111, the separation of scales in slender structures that grow in the plane and out of it links the physical problem of growing elastic films to the geometrical problem of determining a slowly evolving approximately two-dimensional film in three dimensions. Indeed, as we will see, many of the questions we review here are related to a classical theme in differential geometry—that of embedding a shape with a given metric in a space of possibly different dimension 9596, and eventually that of designing the metric to achieve any given shape. However, the goal now is not only to state the conditions when it might be done (or not) but also to determine the resulting shapes in terms of an appropriate mechanical theory and to understand the limiting behaviors of the solutions as a function of the geometric parameters.

The outline of this paper is as follows. Starting from the view that shape is the consequence of metric frustration in an ambient space, in section 2 we describe the background and objectives of non-Euclidean elasticity formalism as well as present an example of growth equations in this context. In section 3 we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films in section 4, where we also offer a new estimate regarding the scaling of non-Euclidean energies from convex integration. In section 5, we focus on the asymptotic behaviors of the solutions as the (scaled) thickness of the films becomes vanishingly small and the local curvature can become large. In section 6 we digress to consider the weak prestrains and the related Monge–Ampère constrained energies. In section 7, the complete range of results is compared with the hierarchy of classical geometrically nonlinear theories for elastic plates and shells without prestrain. Along the way and particularly in section 8, we discuss open mathematical problems and future research directions.

2. Non-Euclidean elasticity and an example of growth equations

An inexpensive surgical experiment serves as a clue to the biological processes at work in determining shape: if one takes a sharp knife and cuts a long, rippled leaf into narrow strips parallel to the midrib, the strips flatten out when “freed” from the constraints of being contiguous with each other. This suggests that the shape is the result of geometric frustration and feedback, driven by the twin effects of embedding a non-Euclidean metric due to inhomogeneous growth and of minimizing an elastic energy that selects the particular observed shape. Experiments confirm the generality of this idea in a variety of situations, ranging from undulating submarine avascular algal blades to saddle-shaped, coiled, or edge-rippled leaves of many terrestrial plants 6689. Understanding the origin of the morphologies of slender structures as a consequence of either their growth or the constraints imposed by external forces, requires a mathematical theory for how shape is generated by inhomogeneous growth in a tissue.

2.1. Non-Euclidean elasticity

Biological growth arises from changes in four fields: cell number, size, shape, and motion, all of which conspire to determine the local metric, which in general will not be compatible with the existence of an isometric immersion. For simplicity, growth is often coarse-grained by averaging over cellular details, thus ignoring microscopic structure due to cellular polarity, orientation (nematic order), anisotropy, etc. While recent work has begun to address these more challenging questions 93122, we limit our review to the case of homogeneous, isotropic thin growing bodies. This has proceeded along three parallel paths, all leading to a set of coupled hyperelliptic PDEs that follow from a variational principle:

•

by using the differential geometry of surfaces as a starting point to determine a plausible class of elastic energies written in terms of the first and second fundamental forms or their discrete analogues and deviations from some natural state 49;

•

by drawing on an analogy between growth and thermoelasticity 91 and plasticity 73, since they both drive changes in the local metric tensor and the second fundamental form, and by using this to build an energy functional whose local minima determines shape;

•

by starting from a three-dimensional theory for a growing elastic body with geometrically incompatible growth tensors, driving the changes of the first and second fundamental forms of a two-dimensional surface embedded in three dimensions 727.

The resulting shape can be seen as a consequence of the heterogeneous incompatibility of strains that leads to geometric (and energetic) frustration. This coupling between residual strain and shape implies an energetic formulation of non-Euclidean elasticity that attempts to minimize an appropriate energy associated with the frustration between the induced and intrinsic geometries. Within this framework, a few different types of problems may be posed:

•

questions about the nature of the (regular and singular) solutions that arise;

•

questions about their connection to experimental observations;

•

problems related to the limiting behavior of the solutions and their associated energies in the limit of small (scaled) thickness;

•

questions about identifying the form of feedback laws linking growth to shape that lead to the self-regulated reproducible forms seen in nature;

•

problems in formulating inverse problems in the context of shaping sheets for function.

2.2. An example of growth equations

To get a glimpse of the analytical structures to be investigated, we begin by writing down a minimal theory that couples growth to the shape of a thin lamina of uniform thickness 78991, now generalized to account for differential growth:

$$\begin{equation} \begin{split} & \Delta (\operatorname {tr}\boldsymbol{\sigma }) +\frac{\alpha }{2} \det \boldsymbol{\kappa }=- \alpha \Delta (\operatorname {tr} \mathbf{s}),\\ & \beta \Delta (\operatorname {tr}\boldsymbol{\kappa }) - \operatorname {tr} (\boldsymbol{\sigma }\boldsymbol{\kappa })=-\beta \Delta (\operatorname {tr} \mathbf{b}). \end{split} {\tag{2.1}} \end{equation}$$

Here, $\Delta$ is the two-dimensional Laplace–Beltrami operator, $\boldsymbol{\sigma }$ is the two-dimensional depth-averaged stress tensor, and $\boldsymbol{\kappa }$ is the curvature tensor. The scalar coefficients $\alpha$ and $\beta$ characterize the elastic moduli of the sheet, assumed to be made of a linear isotropic material: $\alpha$ is the resistance to stretching (and shearing) in the plane, and $\beta$ is the resistance to bending out of the plane. The right-hand side of 2.1 quantifies the source that drives in-plane differential growth due to a prescribed metric tensor $\mathbf{s}$, and the out-of-plane differential growth gradient across the thickness due to a prescribed second fundamental form (a curvature tensor) $\mathbf{b}$.

The first equation in the system 2.1 corresponds to the incompatibility of the in-plane strain due to both the Gauss curvature and the additional contribution from in-plane differential growth, and it is a geometric compatibility relation. The second equation in system 2.1 is a manifestation of force balance in the out-of-plane direction due to the in-plane stresses in the curved shell and to the growth curvature tensor associated with transverse gradients that leads to an effective normal pressure. We observe that $\beta /\alpha = \mathcal{O}(h^2)$, where $h$ is the thickness of the tissue, so there is a natural small parameter in the problem $a = h/L \ll 1$, where $L$ is the lateral size of the system. The nonlinear hyperelliptic equations 2.1 need to be complemented with an appropriate set of boundary conditions on some combination of the displacements, stresses, and their derivatives. However, it is not even clear if and when it is possible to realize reasonable physical surfaces for arbitrarily prescribed tensors $\mathbf{s}, \mathbf{b}$, and so one must resort to a range of approximate methods to determine the behavior of the solutions in general.

There are two large classes of problems associated with the appearance of fine scales or sharp localized conical features, and they are characterized by two distinguished limits of 2.1. These correspond to the situation when either the in-plane stress is relatively large or when it is relatively small. In the first case, when $| \boldsymbol{\sigma }| \sim \alpha$, i.e., the case where stretching dominates, one can rescale equations 2.1 so that they yield the singularly perturbed limit:

$$\begin{equation} \begin{split} & \Delta (\operatorname {tr}\boldsymbol{\sigma }) +\frac{1}{2} \det \boldsymbol{\kappa }=- \Delta (\operatorname {tr} \mathbf{s}),\\ & a^2\Delta (\operatorname {tr} \boldsymbol{\kappa })- \operatorname {tr} (\boldsymbol{\sigma }\boldsymbol{\kappa })=-a^2 \Delta (\operatorname {tr} \mathbf{b}). \end{split} {\tag{2.2}} \end{equation}$$

As $a^2 \rightarrow 0$, at leading order, the second of the equations above implies $\operatorname {tr} (\boldsymbol{\sigma }\boldsymbol{\kappa }) =0$, which has a simple geometric interpretation. Namely, the stress-scaled mean curvature vanishes, which is an interesting generalization of the Plateau–Lagrange problem for minimal surfaces. Then, system 2.2 describes a finely decorated minimal surface, where wrinkling patterns appear in regions with a sufficiently negative stress.

In the second case, when the in-plane stress is relatively small $|\boldsymbol{\sigma }| \sim \alpha a^2$, i.e., the case when bending dominates, one can rescale 2.1 to obtain a different singularly perturbed limit:

$$\begin{equation} \begin{split} & a^2\Delta (\operatorname {tr}\boldsymbol{\sigma }) +\frac{1}{2} \det \boldsymbol{\kappa }=- \Delta (\operatorname {tr} \mathbf{s}),\\ & \Delta (\operatorname {tr} \boldsymbol{\kappa })- \operatorname {tr}(\boldsymbol{\sigma }\boldsymbol{\kappa })=- \Delta (\operatorname {tr}\mathbf{b}). \end{split} {\tag{2.3}} \end{equation}$$

As $a^2 \to 0$, at leading order the first of the equations above yields $\det \boldsymbol{\kappa }=- 2 \Delta (\operatorname {tr}\mathbf{s})$, which can be seen as a Monge–Ampère type equation for the Gauss curvature. Then, system 2.3 describes a spontaneously crumpled, freely growing sheet with conical and ridge-like singularities, similar to the result of many a failed calculation that ends up in the recycling bin.

Adding the growth terms in 2.1 is however only part of the biological picture, since in general there is likely to be feedback; i.e., just as growth leads to shape, shape (and residual strain) can change the growth patterns. Then, the growth tensors $\mathbf{s}, \mathbf{b}$ must themselves be coupled to the shape of the sheet via additional (dynamical) equations.

Open Problem 2.1.

The above description follows the one-way coupling of growth to shape and ignores the feedback from the residual strain. It is known that biological mechanisms inhibit cell growth if the cell experiences sufficient external pressure. Although there are proposals for how shape couples back to growth, this remains a largely open question of much current interest in biology, and we will return to this briefly in the concluding sections.

3. Shape from geometric frustration in growing laminae

The variety of forms seen in the three-dimensional shapes of leaves or flowers, reflects their developmental and evolutionary history and the physical processes that shape them, posing many questions at the nexus of biology, physics, and mathematics. From a biological perspective, it is known that genetic mutants responsible for differential cell proliferation lead to a range of leaf shapes 97125. From a physical perspective, stresses induced by external loads lead to phenotypic plasticity in algal blades that switch between long, narrow blade-like shapes in rapid flow to broader undulating shapes in slow flow 66. Similar questions arise from observations of a blooming flower—long an inspiration for art and poetry, but seemingly not so from scientific perspectives. When a flower blossoms, its petals change curvature on a time scale of a few hours, consistent with the idea that these movements are driven by cellular processes. In flowers that bloom once, differential cell proliferation is the dominant mode of growth, while in those that open and close repeatedly, cell elongation plays an important role.

Although proposed explanations for petal movements posit a difference in growth rate of its two sides (surfaces) or an active role for the midribs, experimental, theoretical, and computational studies 90 have shown that the change of the shape of a lamina is due to excess growth of the margins relative to the center (see Figure 1). Indeed, there is now ample evidence of how relative growth leads to variations in shape in such contexts as leaves, flowers, micro-organisms (i.e., euglenids), swelling sheets of gels, 4d printed structures etc. 3836476263107124. A particularly striking example is that of the formation of self-similar wrinkled structures as shown in the example of a kale leaf in Figure 2. A demonstration of the same phenomenon with everyday materials is also shown in Figure 2—when a garbage bag is torn, its edge shows multiple generations of wrinkles 106.

3.1. The setup

The experimental observations described above suggest a common mathematical framework for understanding the origin of shape: an elastic three-dimensional body $\Omega$ seeks to realize a configuration with a prescribed Riemann metric $g$ by means of an isometric immersion. The deviation from or inability to reach such a state, is due to a combination of geometric incompatibility and the requirements of elastic energy minimization. Borrowing from the theory of finite plasticity 73, where a multiplicative decomposition of the deformation gradient into an elastic and a plastic use was postulated, a similar hypothesis was suggested for growth 104, with the underlying hypothesis of the presence of a reference configuration $\Omega$ with respect to which all displacements are measured.

Let $g:\Omega \to \mathbb{R}^{3\times 3}_{\operatorname {sym}, >}$ be a smooth Romanian metric, given on an open, bounded domain $\Omega \subset \mathbb{R}^3$, and let $u:\Omega \to \mathbb{R}^3$ be an immersion that corresponds to the elastic body. Excluding nonphysical deformations that change the orientation in any neighborhood of the immersion, a natural way to pose the question of the origin of shape is by postulating that it arises from a variational principle that minimizes an elastic energy $\mathcal{E}(u)$ which measures how far a given $u$ is from being an orientation-preserving realization of $g$. Equivalently, $\mathcal{E}(u)$ quantifies the total point-wise deviation of $\nabla u$ from $g^{1/2}$, modulo orientation-preserving rotations that do not cost any energy. The infamy of $\mathcal{E}$ in absence of any forces or boundary conditions is then indeed strictly positive for a non-Euclidean $g$, pointing to existence of residual strain.

Since the matrix $g(x)$ is symmetric and positive definite, it possesses a unique symmetric, positive definite square root $A(x)={g(x)}^{1/2}\in \mathbb{R}^{3\times 3}_{\operatorname {sym}, >}$ which corresponds to the growth prestrain. This allows us to define an energy,

$$\begin{equation} \mathcal{E}(u) = \int _{\Omega } W\left((\nabla u)A^{-1}\right)\nobreakspace \mathrm{d}x \qquad \forall u\in W^{1,2}(\Omega ,\mathbb{R}^3), {\tag{3.1}} \end{equation}$$

where the energy density $W:\mathbb{R}^{3\times 3}\to [0,\infty ]$ obeys the principles of material frame invariance (with respect to the special orthogonal group of proper rotations $SO(3)$), normalization, nondegeneracy, and material consistency valid for all $F\in \mathbb{R}^{3\times 3}$ and all $R\in SO(3)$,

$$\begin{equation} \begin{split} & W(RF) = W(F), \qquad W(\operatorname {Id}_3) = 0, \qquad W(F) \geq c\nobreakspace \operatorname {dist}^2(F, SO(3)), \\ & W(F)\to +\infty \quad \text{as } \det F\to 0+, \quad \text{ and}\quad \forall \det F\leq 0 \quad W(F) = +\infty . \end{split} {\tag{3.2}} \end{equation}$$

These models,⁠¹ corresponding to a range of hyperelastic energy functionals that approximate the behavior of a large class of elastomeric materials, are consistent with microscopic derivations based on statistical mechanics, and they naturally reduce to classical linear elasticity when $|(F^TF)^{1/2} - \operatorname {Id}| \ll 1$. Minimizing the energy 3.1 is thus a prescription for shape and may be defined naturally in terms of the energetic cost of deviating from an isometric immersion.

¹

Examples of $W$ satisfying these conditions are $W_1(F) = |(F^TF)^{1/2} - \operatorname {Id}|^2 + |\log \det F|^q$ or $W_2(F)=|(F^TF)^{1/2} - \operatorname {Id}|^2 + \left|({\det F})^{-1} - 1\right|^q$ for $\det F>0$, where $q>1$ and $W_{1,2}$ equal $+\infty$ if $\det F\leq 0$.

✖

3.2. Isometric immersions and residual stress

The model in 3.1 assumes that the $3$d elastic body $\Omega$ seeks to realize a configuration with a prescribed Riemannian metric $g$, through minimizing the energy that is determined by the elastic part $F_e=(\nabla u)A^{-1}$ of the deformation gradient $\nabla u$. Observe that $W(F_e)=0$ if and only if $F_e\in SO(3)$ in $\omega$, or equivalently when:

$$\begin{equation*} (\nabla u)^T\nabla u = g\quad \text{ and } \quad \det \nabla u > 0 \quad \text{in }\;\omega , \end{equation*}$$

Further, any $u\in W^{1,2}(\Omega ,\mathbb{R}^3)$ that satisfies the above must automatically be smooth. Indeed, writing $\nabla u = Rg^{1/2}$ for some rotation field $R:\Omega \to SO(3)$, it follows that $u\in W^{1,\infty }$ and so $\operatorname {div}\big (\operatorname {cof}\,\nabla u\big )=0$ holds, in the sense of distributions.⁠² Further, we have

²

The divergence of a matrix field is taken row-wise.

✖

$$\begin{equation} \begin{split} \operatorname {cof}\,\nabla u &=\operatorname {cof}(Rg^{1/2})=\det (Rg^{1/2}) \big (Rg^{1/2}\big )^{T, -1}\\ &= \sqrt {\det g}\big (R g^{-1/2}\big )= \sqrt {\det g}\big ((\nabla u) g^{-1}\big ). \end{split}{\tag{3.3}} \end{equation}$$

It follows that each of the three scalar components of $u$ is harmonic with respect to the Laplace–Beltrami operator $\Delta _g$, and thus $u$ must be smooth:

$$\begin{equation*} 0=\operatorname {div}\big (\operatorname {cof}\,\nabla u\big )=\operatorname {div}\big ((\sqrt {\det g})(\nabla u) g^{-1}\big ) = \sqrt {\det g}\cdot \big [\Delta _gu^m\big ]_{m=1}^3. \end{equation*}$$

Thus, $\mathcal{E}(u)=0$ if and only if the deformation $u$ is an orientation-preserving isometric immersion of $g$ into $\mathbb{R}^3$. Such smooth (local) immersion exists 112, Vol. II, Chapter 4 and is automatically unique up to rigid motions of $\mathbb{R}^3$, if and only if the Riemann curvature tensor $[\mathcal{R}_{ij, kl}]_{i,j,k,l=1\cdots 3}$ of $g$ vanishes identically throughout $\Omega$.

It is instructive to point out that one could define the energy as the difference between the prescribed metric $g$ and the pull-back metric of $u$ on $\Omega$:

$$\begin{equation*} I(u)= \int _{\Omega } |(\nabla u)^T\nabla u - g|^2\nobreakspace \mathrm{d}x. \end{equation*}$$

From a variational point of view, the formulation above does not capture an essential aspect of the physics, namely that thin laminae resist bending deformations that are a consequence of the extrinsic geometry, and thus depend on the mean curvature as well. Indeed, the functional $I$ always minimizes to $0$ because there always exists a Lipschitz isometric immersion $u\in W^{1,\infty }(\Omega ,\mathbb{R}^3)$ of $g$, for which $I(u) = 0$. If $\mathcal{R}_{ij,kl}(x)\neq 0$ for some $x\in \Omega$, then such $u$ must have a folding structure 50 around $x$; it cannot be orientation preserving (or reversing) in any open neighborhood of $x$. Perhaps even more surprisingly, the set of such Lipschitz isometric immersions is dense in the set of short immersions as for every $u_0\in \mathcal{C}^1(\bar{\Omega }, \mathbb{R}^3)$ satisfying $(\nabla u_0)^T\nabla u_0<g$,⁠³ there exists a sequence $\{u_n\in W^{1,\infty }(\Omega , \mathbb{R}^3)\}_{n\to \infty }$ satisfying

³

That is, the matrix $g(x) - \nabla u_0(x)^T\nabla u_0(x)$ is positive definite at each $x\in \omega$.

✖

$$\begin{equation*} I(u_n) = 0 \quad \text{ and } \quad \|u_n-u_0\|_{\mathcal{C}(\Omega )} \to 0\quad \text{ as }\; n \to \infty . \end{equation*}$$

The above statement is an example of the $h$-principle in differential geometry, and it follows through the method of convex integration (the Nash–Kuiper scheme), to which we come back in the following sections. An intuitive example in dimension $1$ is shown in Figure 3. Setting $g\equiv 1$ on $\Omega =(-1,1)\subset \mathbb{R}^1$, it is easily seen that any $u_0:(-1,1)\to \mathbb{R}$ with Lipschitz constant less than $1$ can be uniformly approximated by $u_n$ having the form of a zigzag, where $|u_n'|=1$.

Regarding the energy $\mathcal{E}$ in 3.1, in 85 it has been proved that $\inf \mathcal{E}>0$ for any $g$ with no orientation-preserving isometric immersion. This results in the dichotomy: either $g$ and $\mathcal{E}$ are, by a smooth change of variable, equivalent to the case with $g=\operatorname {Id}_3$ and $\min \mathcal{E}=0$, or otherwise the zero energy level cannot be achieved through a sequence of weakly regular $W^{1,2}$ deformations. The latter case points to existence of residual strain at free equilibria.

Proposition 3.1 (85).

If $[\mathcal{R}_{ij,kl}] \not \equiv 0$ in $\Omega$, then $\inf \big \{\mathcal{E(}u);\nobreakspace u\in W^{1,2}(\Omega ,\mathbb{R}^3)\big \}>0$.

Sketch of proof.

Assume, by contradiction, that $\mathcal{E}(u_n)\to 0$ along some sequence $\{u_n\in W^{1,2}(\Omega , \mathbb{R}^3)\}_{n\to \infty }$. By truncation and approximation in Sobolev spaces, we may, without loss of generality, assume that each $u_n$ is Lipschitz with a uniform Lipschitz constant $M$. Decompose $u_n=z_n+w_n$ as a sum of a deformation that is clamped at the boundary,

$$\begin{equation*} [\Delta _g z_n^m]_{m=1}^3 = [\Delta _g u_n^m]_{m=1}^3 \quad \text{in }\Omega , \qquad z_n=0 \quad \text{on } \partial \Omega , \end{equation*}$$

and a harmonic correction, $[\Delta _g w_n^m]_{m=1}^3 = 0$ in $\Omega$, with $w_n=u_n$ on $\partial \Omega$. Observe that

$$\begin{equation*} \begin{split} \int _\Omega \big \langle (\sqrt {\det g})&\nabla z_n g^{-1}:\nabla z_n\big \rangle \;\mathrm{d}x = \int _\Omega \big \langle (\sqrt {\det g})\nabla u_n g^{-1}:\nabla z_n\big \rangle \;\mathrm{d}x \\ & = \int _\Omega \big \langle (\sqrt {\det g})\nabla u_n g^{-1}:\nabla z_n\big \rangle - \langle \operatorname {cof}\,\nabla u_n : \nabla z_n\rangle \;\mathrm{d}x \\ & \leq \|\nabla z_n\|_{L^2} \big \| (\sqrt {\det g})\nabla u_n g^{-1} -\operatorname {cof}\,\nabla u_n\big \|_{L^2}, \end{split} \end{equation*}$$

where the first equality follows by $z_n=0$ on $\partial \Omega$, as $\operatorname {div}\big ((\sqrt {\det g})(\nabla u) g^{-1}\big ) = \sqrt {\det g} \big [\Delta _gu^m\big ]_{m=1}^3$, while the second by $\operatorname {div}(\operatorname {cof}\,\nabla u)=0$. The left-hand side is also equivalent to $\|\nabla z_n\|_{L^2}^2$, so

$$\begin{equation} \|\nabla z_n\|_{L^2} \leq C\big \| (\sqrt {\det g})\nabla u_n g^{-1} -\operatorname {cof}\,\nabla u_n\big \|_{L^2} \leq C_M \mathcal{E}(u_n)^{1/2}\to 0. {\tag{3.4}} \end{equation}$$

Above, we used 3.3 which ensures vanishing of the expression under the norm when $(\nabla u_n)A^{-1}\in SO(3)$, together with Lipschitz continuity of the operator in the integral expression for $\mathcal{E}$. In particular, we get that both sequences $\{\nabla z_n\}_{n\to \infty }$ and $\{\nabla w_n\}_{n\to \infty }$ are bounded in $L^2$.

Since $\{w_n\}$ are harmonic, this further implies that $\nabla u_n$ converges, up to a subsequence, in $L^2_{\operatorname {loc}}$ to some $\nabla u$. Then by 3.4 $\nabla u_n\to \nabla u$, which yields $\mathcal{E}(u) =0$, and ends the proof.

■

Open Problem 3.2.

In the above context, prove that $\inf \mathcal{E}$ as in Proposition 3.1 is equivalent to $\|[\mathcal{R}_{ij, kl}]\|_{H^{-2}}^{2}$, up to multiplicative constants depending on $\Omega$ and $W$ but not on $g$. The case of $\Omega \subset \mathbb{R}^2$ and $\mathcal{R}$ replaced by the Gaussian curvature has been considered in 72.

4. Microstructural patterning of thin elastic prestrained films

Inspired partly by biological observations of growth-induced patterning in thin sheets and the promise of engineering applications, various techniques have been invented for the construction of self-actuating elastic sheets with prescribed target metrics. The materials typically involve the use of gels that respond to pH, humidity, temperature, and other stimuli 115, and that result in the formation of complex controllable shapes (see Figure 4) that include both large-scale buckling and small-scale wrinkling forms.

In one example 64, NIPA monomers with a BIS crosslinker in water and a catalyst, leads to the polymerization of a cross-linked elastic hydrogel, which undergoes a sharp, reversible, volume-reduction transition at a threshold temperature, allowing for temperature-controlled swelling in thin composite sheets. Another method 63 involves the photopatterning of polymer films to yield temperature-responsive gel sheets with the ability to print nearly continuous patterns of swelling. A third method 47 uses 3d printing of complex-fluid based inks to create bilayers with varying line density and anisotropy in order to achieve control over the extent and orientation of swelling. All these methods have been used to fabricate surfaces with constant Gaussian curvature (spherical caps, saddles, cones) or zero mean curvature (Enneper’s surfaces), as well as more complex and nearly closed shapes. A natural question that these controlled experiments raise is the ability (or lack thereof) of the resulting patterns to approximate isometric immersions of prescribed metrics. From a mathematical perspective, this leads to questions of the asymptotic behavior of energy minimizing deformations and their associated energetics.

4.1. The setup

In this and the next sections, we will consider a family $(\Omega ^h, u^h, g, A, \mathcal{E}^h)_{h>0}$ (or more generally $(\Omega ^h, u^h, g^h, A^h, \mathcal{E}^h)_{h>0}$) given in the function of the film’s thickness parameter $h$. The main objective is now to predict the scaling of $\inf \mathcal{E}^h$ as $h\to 0$ and to analyze the asymptotic behavior of minimizing deformations $u^h$ in relation to the curvatures associated with the prestrain $A$. We assume that $A= g^{1/2}:\overline{\Omega ^1}\to \mathbb{R}^{3\times 3}_{\operatorname {sym}, >}$ is a smooth, symmetric, and positive definite tensor field on the unit thickness domain $\Omega ^1$, where for each $h>0$ we define

$$\begin{equation*} \Omega ^h=\omega \times \big (-\frac{h}{2}, \frac{h}{2}\big ). \end{equation*}$$

The open, bounded set $\omega \subset \mathbb{R}^2$ with Lipschitz boundary is viewed as the midplate of the thin film $\Omega ^h$, on which we pose the energy of elastic deformations,

$$\begin{equation} \mathcal{E}^h(u^h) = \frac{1}{h}\int _{\Omega ^h} W\big ((\nabla u^h)A^{-1}\big )\;\mathrm{d}x\qquad \text{ for all }\; u^h\in W^{1,2}(\Omega ^h,\mathbb{R}^3). {\tag{4.1}} \end{equation}$$

4.2. Isometric immersions and energy scaling

As in section 3.2, there is a connection between $\inf \mathcal{E}^h$ and existence of isometric immersions, although this case is a bit more subtle. In the context of dimension reduction, this connection relies on the isometric immersions of the midplate metric $g(\cdot ,0)_{2\times 2}$ on $\omega$ into $\mathbb{R}^3$, namely parametrized surfaces $y:\omega \to \mathbb{R}^3$ with

$$\begin{equation} (\nabla y)^T\nabla y = g(\cdot ,0)_{2\times 2}\quad \text{in } \omega . {\tag{4.2}} \end{equation}$$

It turns out that existence of $y$ with regularity $W^{2,2}$ is equivalent to the vanishing of $\inf \mathcal{E}^h$ of order square in the film’s thickness $h\to 0$. The following result was proved first for $g=g(x')$ in 85 and then in the abstract setting of Riemannian manifolds in 71.

Theorem 4.1 (12).

Let $u^h\in W^{1,2}( \Omega ^h, \mathbb{R}^3)$ satisfy $\mathcal{E}^h(u^h) \le Ch^2$. Then we have:

(i)

Compactness. There exist $c^h\in \mathbb{R}^3$ and $R^h\in SO(3)$ such that the rescaled deformations $y^h(x', x_3) = R^h u^h(x', hx_3) - c^h$ converge, up to a subsequence in $W^{1,2}(\Omega ^1, \mathbb{R}^3)$, to some $y \in W^{2,2}(\Omega ^1, \mathbb{R}^3)$ depending only on the tangential variable $x'$ and satisfying 4.2.

(ii)

Liminf inequality. There holds the lower bound,