Minimizing Configurations for Elastic Surface Energies with Elastic Boundaries

Abstract

We study critical surfaces for a surface energy which contains the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$, coupled to the elastic energy of the boundary curve. We investigate the existence of equilibria with $H\equiv -c_o$. When $c_o \geqslant 0$, we characterize those cases where the infimum of this energy is finite for topological annuli, and we find the minimizer in the cases that it exists. Results for topological disks are also given.

Appendix A: First Variation Formula

In this appendix, we will compute the first variation formula of the total energy E. Most of the calculations below are well known and they are included for completeness.

Let $\delta X$ be a smooth $\mathbf{R }^3$ valued map on $\Sigma$ which we consider as a variation field, i.e., the linear term of a deformation $X_\epsilon :=X+\epsilon \, \delta X+{\mathcal {O}}(\epsilon ^2)$.

Although for functionals with geometric character (i.e., invariant under changes of parameterization) only normal variations are usually considered, for surfaces with boundary it is essential to use variations having normal as well as tangential components. In general, the boundary is not invariant under tangential variations and, hence, computing the first variation formula only for normal variations leads to a loss of valuable information.

We decompose $\delta X$ as $V+\psi \nu$ with V tangent to the surface. At times we will denote with “dot” derivatives with respect to the variation parameter $\epsilon$, i.e., $\delta f={\dot{f}}$ for an $\epsilon$ dependent function f on $\Sigma$.

We first consider the case $V\equiv 0$. A straightforward calculation gives the variation for the normal field as

$$\begin{aligned} {{{\dot{\nu }}}}=-\nabla \psi \,, \end{aligned}$$

where we denote the surface gradient operator by $\nabla$.

We next compute the variation for the components metric tensor $g_{ij}$ with respect to a locally defined frame field $\{e_1, e_2\}$ which is orthonormal for the metric induced by $X_0\equiv X$, i.e., $g_{ij}(0)=\delta _{ij}$,

$$\begin{aligned} {\dot{g}}_{ij}= & {} \delta \left( X_i\cdot X_j\right) ={\dot{X}}_i\cdot X_j+X_i\cdot {\dot{X}}_j=\left( \psi \nu \right) _i\cdot X_j+X_i\cdot \left( \psi \nu \right) _j\\= & {} 2\psi \nu _i\cdot X_j=-2\psi L_{ij}\,, \end{aligned}$$

where $L_{ij}:=-X_i\cdot \nu _j$ are the components of the second fundamental form of the surface. Recall that this tensor is symmetric, i.e., $L_{ij}=L_{ji}$.

The induced surface measure on $\Sigma$ is given by $d\Sigma =g^{1/2}\left( e_1^*\wedge e_2^*\right)$, where $g:=\det (g_{ij})$ and $\{e_1^*,e_2^*\}$ is the dual basis. From this and the calculation above, one easily obtains

$$\begin{aligned} \delta d\Sigma =-2H\psi d\Sigma \,. \end{aligned}$$

We now consider the variation for the second fundamental form of the surface $\left( L_{ij}\right)$:

$$\begin{aligned} {\dot{L}}_{ij}= & {} - \delta \left( X_i\cdot \nu _j\right) =- {\dot{X}}_i\cdot \nu _j-X_i\cdot {{{\dot{\nu }}}}_j=-\left( \psi \nu \right) _i\cdot \nu _j-X_i\cdot \left( -\nabla \psi \right) _j\\= & {} -\psi \nu _i\cdot \nu _j+\psi _{,ij}=-\psi L_{ik}L_{kj}+\psi _{,ij}\,. \end{aligned}$$

From the previous two calculations, we obtain, using ${\dot{g}}^{ij}=-{\dot{g}}_{ij}$,

$$\begin{aligned} {\dot{H}}= & {} \delta \left( \frac{1}{2}g^{ij}L_{ij}\right) = \frac{1}{2}\left( {\dot{g}}^{ij}L_{ij}+g^{ij}{\dot{L}}_{ij}\right) \\= & {} \frac{1}{2}\left( 2\psi L_{ij}L_{ij}+g^{ij}\psi _{,ij}-\psi L_{ij}L_{ij}\right) =\frac{1}{2}\left( \Delta \psi +\Vert d\nu \Vert ^2\psi \right) , \end{aligned}$$

where $\Vert d\nu \Vert ^2=L_{ij}L_{ij}=4H^2-2K$ is the square of the norm of the second fundamental form.

In order to compute the pointwise variation for the Gaussian curvature K, we choose the frame so that, in addition to being orthonormal, $L_{ij}=k_i\delta _{ij}$ holds. Here, $k_1$ and $k_2$ are the principal curvatures of the surface. Then we obtain

$$\begin{aligned} {\dot{K}}= & {} \delta \left( \frac{L_{11}L_{22}-L_{12}^2}{g}\right) ={\dot{L}}_{11}k_2+k_1{\dot{L}}_{22}-{\dot{g}}K\\= & {} k_2\psi _{,11}+k_1\psi _{,22}-\left( k_2k_1^2+k_1k_2^2\right) \psi +4HK\psi \\= & {} k_2\psi _{,11}+k_1\psi _{,22}+2HK\psi \,. \end{aligned}$$

The Codazzi equations with respect to this frame are:

$$\begin{aligned} e_2(k_1)=:k_{1,2}= & {} -\left( k_1-k_2\right) \nabla _1e_2\cdot e_1\,,\\ e_1(k_2)=:k_{2,1}= & {} \left( k_1-k_2\right) \nabla _2e_1\cdot e_2\,. \end{aligned}$$

Define $A:=\left( d\nu +2H\,Id\right) \nabla \psi = k_2\psi _{1}e_1+k_1\psi _{2}e_2$. Below we use the notation $\psi _{i,j}$ to denote $e_j(\psi _i)$ and as before $\psi _{,ij}$ denotes the (i, j) component of the Hessian tensor. Using the Codazzi equations, we have that the divergence $\nabla \cdot$ of A is given by

$$\begin{aligned} \nabla \cdot A= & {} \nabla _i\left( k_2\psi _{1}e_1+k_1\psi _{2}e_2\right) \cdot e_i\\= & {} \left( k_2\psi _1\right) _1+\left( k_1\psi _2\right) _2+k_2\psi _1\nabla _2e_1\cdot e_2+k_1\psi _2\nabla _1e_2\cdot e_1\\= & {} k_{2,1}\psi _1+k_2\psi _{1,1}+k_{1,2}\psi _2+k_1\psi _{2,2}+k_2\psi _1\nabla _2e_1\cdot e_2+k_1\psi _2\nabla _1e_2\cdot e_1\\= & {} \psi _1\left( k_1-k_2\right) \nabla _2e_1\cdot e_2+k_2\psi _{1,1}-\psi _2\left( k_1-k_2\right) \nabla _1e_2\cdot e_1\\&+k_1\psi _{2,2}+k_2\psi _1\nabla _2e_1\cdot e_2+k_1\psi _2\nabla _1e_2\cdot e_1\\= & {} \psi _1k_1\nabla _2e_1\cdot e_2+k_2\psi _{1,1}+\psi _2k_2\nabla _1e_2\cdot e_1+k_1\psi _{2,2}\\= & {} k_1\psi _{,22}+k_2\psi _{,11}\,. \end{aligned}$$

Comparing this with the expression for ${\dot{K}}$ above, we obtain

$$\begin{aligned} {\dot{K}}=\nabla \cdot \left( \left[ d\nu +2H\, Id\right] \nabla \psi \right) +2HK\psi \,. \end{aligned}$$

For the case of a variation tangential to the surface, i.e., $\delta X=V$, it is clear that ${\dot{H}}=\nabla H\cdot V$, $\dot{K}=\nabla K\cdot V$ and $\delta d\Sigma =\left( \nabla \cdot V\right) d\Sigma$, where $\nabla \cdot V$ denotes the divergence of V.

We are now in a position to compute the variation for the Helfrich energy ( Helfrich (1973))

$$\begin{aligned} {\mathcal {H}}[\Sigma ]=\int _\Sigma a\left( H+c_o\right) ^2\,\mathrm{d}\Sigma \,. \end{aligned}$$

For $\delta X=\psi \nu +V$, integrating by parts we obtain

$$\begin{aligned} \delta {\mathcal {H}}[\Sigma ]= & {} \int _\Sigma 2\left( H+c_o\right) {\dot{H}}\,\mathrm{d}\Sigma +\int _\Sigma \left( H+c_o\right) ^2\delta (\mathrm{d}\Sigma )\\= & {} \int _\Sigma 2\left( H+c_o\right) \left( \frac{1}{2}\left[ \Delta \psi +(4H^2-2K)^2\psi \right] +\nabla H\cdot V \right) \mathrm{d}\Sigma \\&+\int _\Sigma \left( H+c_o\right) ^2 \left( -2H\psi +\nabla \cdot V\right) \mathrm{d}\Sigma \\= & {} \int _\Sigma \left( H+c_o\right) \left( \Delta \psi +2(H^2-K-c_oH)\psi \right) +\nabla \cdot \left[ (H+c_o)^2V\right] \mathrm{d}\Sigma \\= & {} \int _\Sigma \left[ \Delta (H+c_o)+2(H+c_o)(H^2-K-c_oH) \right] \psi \,\mathrm{d}\Sigma \\&+\oint _{\partial \Sigma } \left( H+c_o\right) \partial _n\psi - \partial _n\left( H+c_o\right) \psi +\left( H+c_o\right) ^2V\cdot n\,\mathrm{d}s\,. \end{aligned}$$

In the last line, we have used Green’s Second Identity.

Similarly, for $\delta X=\psi \nu +V$ the variation for the total Gaussian curvature is given by

$$\begin{aligned} \delta \left( \int _\Sigma K\,\mathrm{d}\Sigma \right)= & {} \int _\Sigma {\dot{K}}\,\mathrm{d}\Sigma +K\,\delta (\mathrm{d}\Sigma )\\= & {} \int _\Sigma \left( \nabla \cdot \left( \left[ \mathrm{d}\nu +2H\, Id\right] \nabla \psi \right) +\nabla K\cdot V+K\nabla \cdot V\right) d\Sigma \\= & {} \oint _{\partial \Sigma } \left( \left[ d\nu +2H\, Id\right] \nabla \psi +K V\right) \cdot n\,\mathrm{d}s\\= & {} \oint _{\partial \Sigma } \left( \kappa _n\partial _n\psi -\tau _g\psi '+K V\cdot n\right) \mathrm{d}s\,, \end{aligned}$$

where in the third line we have used the Divergence Theorem.

Finally, to compute the first variation formula of the bending energy of the boundary we need to know the pointwise variation of the squared curvature. For an arbitrary parameter t, we have

$$\begin{aligned} \delta \left( \kappa ^2\right)= & {} \delta \left( T'\cdot T'\right) =\delta \left( \frac{T_t}{\Vert C_t\Vert }\cdot \frac{T_t}{\Vert C_t\Vert }\right) =2\delta \left( \frac{T_t}{\Vert C_t\Vert }\right) \cdot T'\\= & {} 2\left( {\dot{T}}\right) '\cdot T'-2\kappa ^2 T\cdot \left( {\dot{C}}\right) '=2T' \cdot \left( {\dot{C}}\right) ''-4\kappa ^2 T\cdot \left( {\dot{C}}\right) ', \end{aligned}$$

since ${\dot{T}}=\left[ \left( {\dot{C}}\right) '\right] ^\perp$ and, hence, $\left( {\dot{T}}\right) '=\left( {\dot{C}}\right) ''-T'\cdot \left( {\dot{C}}\right) '-T\cdot \left( {\dot{C}}\right) ''$. Next, from the induced measure on $\partial \Sigma$, we directly obtain $\delta (ds)=T\cdot \left( {\dot{C}}\right) 'ds$. Combining both things we get

$$\begin{aligned} \delta \left( \oint _{\partial \Sigma }\left[ \alpha \kappa ^2+\beta \right] ds\right)= & {} \oint _{\partial \Sigma } \alpha \delta \left( \kappa ^2\right) ds+\left( \alpha \kappa ^2+\beta \right) \delta (\mathrm{d}s)\\= & {} \oint _{\partial \Sigma } \left( 2\alpha T'\cdot \left[ {\dot{C}}\right] ''-\left[ 3\alpha \kappa ^2-\beta \right] T\cdot \left[ {\dot{C}}\right] '\right) \mathrm{d}s\\= & {} \oint _{\partial \Sigma }\left( 2\alpha T'''+\left[ (3\alpha \kappa ^2-\beta )T\right] '\right) \cdot {\dot{C}}\,\mathrm{d}s\,, \end{aligned}$$

where in the last equality we have integrated by parts.

The same variations have been obtained using different techniques in Biria et al. (2013) and Tu and Ou-Yang (2004), to mention a couple. For the boundary energy see also Appendix of Langer and Singer (1986).