Removing the center-pinning condition $u(0)=0$ in the $\Gamma$-convergence derivation of the d-cone obstacle problem for thin elastic sheets

The developable cone ('d-cone') is the shape of a thin elastic sheet pressed at its center into a hollow cylinder. Figalli and Mooney (2018) proved that the nonlinear membrane-bending plate energies of a sheet of thickness $h$, normalized by $h^2|\log h|$, $\Gamma$-converge as $h\to0$ on the class $X=\{u\in W^{1,4}(B_1;\mathbb R^3):u(0)=0,\ u|_{S^1}\in\mathbb S^2\}$ to the bending energy of one-homogeneous isometries, and used this to derive and solve a fourth-order obstacle problem for curves on $\mathbb S^2$ describing the d-cone, rigorously confirming the physics prediction of a single fold whose opening angle is independent of the pressing depth. The condition $u(0)=0$, which pins the sheet's center at the cone tip, is imposed rather than derived. In Remark 2.6 of the published version the authors state that removing it 'using smallness of the membrane energy' is an interesting open problem, noting that the only known route (Brandman-Kohn-Nguyen) requires boundary data that cross every great circle of $\mathbb S^2$ and 'is impossible for curves of length $2\pi$' - exactly the class on which the $\Gamma$-limit is finite. The problem is to prove, or disprove, that the same $\Gamma$-convergence holds on the constraint-free space $\tilde X=\{u\in W^{1,4}(B_1;\mathbb R^3):u|_{S^1}\in\mathbb S^2\}$ (and on its obstacle-constrained analogue, the physical d-cone setting), with the compactness and lower bound obtained from smallness of the membrane energy alone.

Background

If the center of a thin elastic sheet resting on top of a hollow cylinder is pressed down by a small distance $\varepsilon$, the sheet takes an almost conical shape called a developable cone, or 'd-cone' ('developable' means the deformed sheet can be isometrically flattened back into the plane, stretching being avoided). Experiments show that the sheet stays in contact with the cylinder except over one single interval - it has 'one fold' - and that the angle subtended by this fold is independent of $\varepsilon$ for small $\varepsilon$ (Cerda-Mahadevan).

A standard two-dimensional model for a sheet of thickness $h>0$ is the nonlinear plate energy $E_h(u)=h^2\int_{B_1}|D^2u|^2\,dx+\int_{B_1}\mathrm{dist}^2(Du,O(2,3))\,dx,$ where $B_1\subset\mathbb R^2$ is the unit disk (the reference sheet), $u:B_1\to\mathbb R^3$ is the deformation, $O(2,3)=\{P\in\mathbb R^{3\times2}:P^TP=I_2\}$ is the set of linear isometries of $\mathbb R^2$ into $\mathbb R^3$, and $D^2u$ is the Hessian. The first ('bending') term penalizes curvature and the second ('membrane') term penalizes local stretching. For boundary data describing a d-cone the minimal energy scales like $h^2|\log h|$ (Brandman-Kohn-Nguyen; Müller-Olbermann), so one studies the normalized energies $\hat E_h:=E_h/(h^2|\log h|)$.

$\Gamma$-convergence is the notion of variational convergence appropriate for passing to limits in minimization problems: $\hat E_h$ $\Gamma$-converges to $E^0$ on a topological space if every sequence converging in that space satisfies the lower bound $\liminf_{h\to0}\hat E_h(u_h)\ge E^0(u)$, and if for each admissible $u$ there is a recovery sequence $u_h\to u$ with $\lim_{h\to0}\hat E_h(u_h)=E^0(u)$; it implies convergence of (almost-)minimizers.

Figalli and Mooney proved that $\hat E_h$ $\Gamma$-converges, as $h\to0$, on the class $X=\{u\in W^{1,4}(B_1;\mathbb R^3):u(0)=0,\ u|_{S^1}\in\mathbb S^2\}$ with the $W^{1,4}$ topology (natural because $W^{1,4}$ convergence implies convergence of the membrane energy), to the functional $E^0$ that equals $\frac{1}{\log 2}\int_{B_1\setminus B_{1/2}}|D^2u|^2\,dx$ on one-homogeneous isometries - maps of the form $u(r,\theta)=r\gamma(\theta)$ with $\gamma$ unit-speed, which are exactly the developable cones and automatically satisfy $u(0)=0$ with boundary curves of length $2\pi$ - and $+\infty$ elsewhere. To model the pressed sheet they impose the obstacle $O_\varepsilon=\{x_1^2+x_2^2>1-\varepsilon^2\}\cap\{x_3<\varepsilon\}$ (the exterior of the cylinder below its rim) and restrict to $X_\varepsilon=X\cap\{u(B_1)\cap O_\varepsilon=\emptyset\}$; the same $\Gamma$-convergence holds there. Minimizing the limit energy over $X_\varepsilon$ is then equivalent to the fourth-order obstacle problem of minimizing the elastica-type bending energy $F(\gamma)=\int_{S^1}\kappa^2\,ds$ over unit-speed closed curves $\gamma$ on $\mathbb S^2$ with $\gamma\subset\{x_3\ge\varepsilon\}$, where $\kappa$ is the geodesic curvature of $\gamma$ (the norm of the tangential component of $\gamma''$; on the unit sphere $\kappa^2=|\gamma''+\gamma|^2$). The problem is called fourth-order because the Euler-Lagrange variational inequality $\kappa''+((1+\lambda)+\kappa^2/2)\kappa\ge0$ involves four derivatives of $\gamma$, in contrast with the classical second-order scalar obstacle problem. Figalli and Mooney prove that for all small $\varepsilon$ the minimizer lifts off the obstacle on exactly one interval $I_\varepsilon$ with $|I_\varepsilon|\to\ell_0\in(2.42,2.43)$, rigorously justifying the physics predictions (the passage $\varepsilon\to0$ to a linearized one-dimensional obstacle problem had been established earlier by Olbermann).

The hypothesis $u(0)=0$ pins the center of the sheet to the tip of the cone; it is an artificial constraint, imposed rather than derived. It enters the proof of the liminf inequality: the geometric estimates showing that low normalized energy forces the boundary data to converge to a unit-speed curve and the maps to their cones rely on this anchoring. The scaling analysis of Brandman-Kohn-Nguyen similarly assumes, rather than proves, that the displacement near the sheet's center is of order $h|\log h|$; their Remark 2 observes that for boundary data crossing every great circle of $\mathbb S^2$ such control can instead be derived from smallness of the membrane energy. In Remark 2.6 of the published version, Figalli and Mooney state verbatim: 'It would be interesting to remove the condition $u(0)=0$ using smallness of the membrane energy. As observed in [BKN] (see Remark 2), one could do this for boundary data that cross every great circle, but this is impossible for curves of length $2\pi$.' Length-$2\pi$ curves are exactly the boundary data of one-homogeneous isometries - the class on which the $\Gamma$-limit is finite - and the boundary curves of the obstacle problem, confined to $\{x_3\ge\varepsilon\}$ and thus missing the equator, can never cross every great circle. The known route therefore fails precisely where it is needed, and the removal of $u(0)=0$ is open.

Problem Statement

Prove, or disprove, that the $\Gamma$-convergence result of Figalli-Mooney remains valid once the artificial constraint $u(0)=0$ is removed.

Let $\tilde X:=\{u\in W^{1,4}(B_1;\mathbb R^3):u|_{S^1}\in\mathbb S^2\}$ with the $W^{1,4}$ topology, let $\hat E_h:=E_h/(h^2|\log h|)$ (where $E_h(u)=h^2\int_{B_1}|D^2u|^2\,dx+\int_{B_1}\mathrm{dist}^2(Du,O(2,3))\,dx$ is the nonlinear plate energy of thickness $h$, $O(2,3)$ denoting the set of linear isometries of $\mathbb R^2$ into $\mathbb R^3$) be extended by $+\infty$ on $\tilde X\setminus W^{2,2}(B_1;\mathbb R^3)$, and define $\tilde E^0$ on $\tilde X$ by $\tilde E^0(u)=\frac{1}{\log 2}\int_{B_1\setminus B_{1/2}}|D^2u|^2\,dx$ if $u\in W^{2,2}(B_1\setminus B_{1/2})$ is a one-homogeneous isometry, i.e. of the form $u(r,\theta)=r\gamma(\theta)$ with $\gamma:S^1\to\mathbb S^2$ unit-speed (such $u$ automatically satisfy $u(0)=0$ and have unit-speed boundary curves of length $2\pi$), and $\tilde E^0(u)=+\infty$ otherwise.

Question: do the functionals $\hat E_h$ $\Gamma$-converge on $\tilde X$ to $\tilde E^0$ as $h\to0$? The essential new content is the liminf (compactness) inequality: every sequence $u_h\in\tilde X$ with $\sup_h\hat E_h(u_h)<\infty$ must have a subsequence converging in $W^{1,4}$ to a one-homogeneous isometry. The recovery-sequence inequality is essentially unchanged from Figalli-Mooney, since their recovery sequence $u_h=h\,f(r/h)\gamma_0(\theta)$, with $f$ smooth even satisfying $f(s)=|s|$ for $|s|\ge1$ and $f(s)=s^2$ for $|s|\le1/2$, already satisfies $u_h(0)=0$. The physically meaningful version, at which Remark 2.6 is aimed, is the same statement on the obstacle-constrained space $\tilde X_\varepsilon=\tilde X\cap\{u(B_1)\cap O_\varepsilon=\emptyset\}$, where $O_\varepsilon=\{x\in\mathbb R^3:x_1^2+x_2^2>1-\varepsilon^2,\ x_3<\varepsilon\}$ is the region outside the cylinder and below its rim (the d-cone setting, where the boundary curves lie in $\{x_3\ge\varepsilon\}$ and hence cannot cross every great circle); it suffices to establish the result on the unconstrained space $\tilde X$; the obstacle-constrained statement then follows by the same closedness and recovery argument as in Figalli-Mooney's Remark 2.5.

A valid proof must derive the compactness from smallness of the membrane energy $\int_{B_1}\mathrm{dist}^2(Du_h,O(2,3))\,dx$ alone, with no hypothesis that the boundary data cross every great circle of $\mathbb S^2$ - the only known route of this type (Brandman-Kohn-Nguyen, Remark 2) being impossible for the length-$2\pi$ boundary curves on which the $\Gamma$-limit is finite. A disproof consists of an explicit sequence $u_h\in\tilde X$ with $\liminf_{h\to0}\hat E_h(u_h)<\infty$ whose $W^{1,4}$ limit is not a one-homogeneous isometry, or of an identification of the $\Gamma$-limit on $\tilde X$ as different from $\tilde E^0$.

References

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