Elastic membranes spanning deformable curves (Q6577061)

The authors consider a closed curve \(\gamma \) in \(\mathbb{R}^{3}\) with curvature \(\kappa \) and torsion \(\tau \), the unit disc \(D\) in \(\mathbb{R}^{2} \) and a parametrization \(X:D\rightarrow \mathbb{R}^{3}\) of a membrane spanning \(\gamma \). They introduce a moving orthonormal frame \(\{t,n,b\}\in W^{1,p}((0,2\pi );SO(3))\), with \(p>1\), which generates a curve \(r\) by integration and define the energy in terms of the frame as \(\mathcal{E} _{f}[(t\mid n\mid b),X]=\int_{0}^{2\pi }f(\kappa ,\tau )d\vartheta +\frac{ \mu }{2}\int_{D}\Psi (\nabla X)dudv\), where \(\kappa =t^{\prime }\cdot n\) and \(\tau =n^{\prime }\cdot b\) play the role of weak signed curvature and weak torsion respectively. The main result of the paper proves that if \(f\) is a convex function which satisfies \(f(a,b)\geq c_{1}\left\vert a\right\vert ^{p}+c_{2}\left\vert b\right\vert ^{p}+c_{3}\), for some \(c_{1}\), \(c_{2}>0\) and \(c_{3}\in \mathbb{R}\) and \(p>1\), and \(\Psi \) is continuous, quasiconvex and satisfies \(c_{4}\left\vert A\right\vert ^{q}\leq \Psi (A)\leq c_{5}\left\vert A\right\vert ^{q}\), for some \(c_{4}\), \(c_{5}>0\) and \(q>1\), then \(\mathcal{E}_{f}\) has a minimizer. Moreover, if \(f\) is \(C^{1}\) and is such that there is \(\alpha >0\) such that \(\left\vert f_{a}(a,b)\right\vert \leq \alpha (1+\left\vert a\right\vert ^{p-1}+\left\vert b\right\vert ^{p-1}) \), \(\left\vert f_{b}(a,b)\right\vert \leq \alpha (1+\left\vert a\right\vert ^{p-1}+\left\vert b\right\vert ^{p-1})\), and if there is \(\beta >0\) such that \(\left\vert D\Psi (A)\right\vert \leq \beta (1+\left\vert A\right\vert ^{2})\), \(\forall A\in \mathbb{R}^{3\times 2}\), then for each minimizer there exist \(\omega \in L^{p}((0,2\pi ))\), \(\chi \in L^{2}((0,2\pi );\mathbb{R} ^{3})\) and \(\lambda \in \mathbb{R}^{3}\) such that first-order necessary conditions can be written which hold a.e. \((u,v)\in D\) and \(\vartheta \in \lbrack 0,2\pi ]\). For the proof, the authors introduce a parametrized curve approach defining the set of constraints by \(\mathcal{C}_{p}=\{(r,X)\in \mathcal{W}_{p}=W^{2,2}((0,2\pi );\mathbb{R}^{3})\times W^{1,2}(D;\mathbb{R} ^{3}):r(0)=r(2\pi )=x_{0}\), \(r^{\prime }(0)=r^{\prime }(2\pi )\), \(\left\vert r^{\prime }\right\vert =1\) on \([0,2\pi ]\), \(\widehat{X}\circ c=r\}\), and the energy functional \(\mathcal{E}_{p}[r,X]=\int_{0}^{2\pi }r^{\prime \prime }d\vartheta +\frac{\mu }{2}\int_{D}\mathfrak{C}\nabla X:\nabla Xdudv\), where \(r\in W^{2,2}((0,2\pi );\mathbb{R}^{3})\) is a \(C^{1}\)-periodic curve clamped at a fixed point and parametrized by arc-length, \(\mu >0\) is the shear modulus, \(X\in W^{2,2}((0,2\pi );\mathbb{R}^{3})\) is the disc-type parametrization of the membrane, \(\mathfrak{C}\in H^{1}(D;\mathcal{L}( \mathbb{R}^{3\times 2};\mathbb{R}^{3\times 2}))\) is the fourth order linear elasticity tensor which should depend on the point \(p\in D\) and satisfies \( \alpha \left\Vert \zeta \right\Vert ^{2}\leq \mathfrak{C}(u,v)\zeta :\zeta \leq \beta (1+\left\Vert \zeta \right\Vert ^{2})\), \(\forall (u,v)\in D\), \( \forall \zeta \in \left\Vert \zeta \right\Vert ^{2}\), for some \(\alpha ,\beta >0\) independent of \((u,v)\in D\). The authors prove that the functional \(\mathcal{E}_{p}\) admits minimizers that satisfy the following first-order necessary conditions \(div(\mathfrak{C}\nabla X)=0\), on \(D\), \( 2r^{\prime \prime \prime }+2\left\vert r^{\prime \prime }\right\vert ^{2}r^{\prime }+\mu F_{\perp }=0\), on \([0,2\pi ]\), where \(F(\vartheta )=\int_{0}^{\vartheta }(\mathfrak{C}\nabla X\circ c)\nu _{D}d\vartheta \), \( \nu _{D}\) is the outer normal of the disc \(D\) and the subscript \(\perp \) stands for the orthogonal part with respect to \(r^{\prime }\). Introducing a penalization term \(\lambda \int_{0}^{2\pi }\left\vert \widehat{X} oc-r\right\vert ^{2}d\vartheta \), with \(\lambda >0\), they prove that for every \(\lambda >0\), the penalized functional \(\mathcal{E}_{p}^{\lambda }\) has a minimizer, introducing a minimizing sequence. They then let \(\lambda \) go to 0. To finally prove the existence of a minimizer to \(\mathcal{E}_{f}\), they prove the closure of the constraint set \(\mathcal{C}_{p}\) and the compactness of sequences of equi-boundeness energy, and they use a lower semicontinuity result. The paper ends with the presentation of a discretization of the problem and of numerical simulations