## Local existence and uniqueness for the $$n$$-dimensional Helfrich flow as a projected gradient flow.(English)Zbl 1247.53083

Let $$\Sigma$$ be a compact, closed, immersed, orientable hypersurface in $${\mathbb R}^{n+1}$$ with mean curvature $$H(x)$$ at $$x\in\Sigma$$. Let $$\nu$$ be a unit normal vector field on $$\Sigma$$. Define the three functionals $$\mathcal W, \mathcal A,\mathcal V$$ by ${\mathcal W}(\Sigma) = \frac{n}{2} \int_\Sigma (H-c_0)^2 \, dS\,, {\;\;} {\mathcal A}(\Sigma) = \int_\Sigma dS\,, {\;\;} {\mathcal V}(\Sigma) = -\,\frac{1}{n+1} \int_\Sigma f\cdot \nu\,dS\,,$ where $$dS$$ is the surface element, $$f$$ is the position vector, and $$c_0$$ is a given constant. The Helfrich variational problem is to find the critical points of $$\mathcal W$$ subject to constraints $${\mathcal A} = {\mathcal A}_0$$ and $${\mathcal V} = {\mathcal V}_0$$.
In the paper under review, the authors consider the Helfrich flow, $$\{ \Sigma(t) \}_{t\geq 0}$$, associated with the Helfrich variational problem. The equation for the flow is $V(t) = - \delta {\mathcal W}(\Sigma(t)) - \lambda_1(\Sigma(t))\, \delta{\mathcal A}(\Sigma(t)) - \lambda_2(\Sigma(t))\, \delta{\mathcal V}(\Sigma(t)) \,,$ where $$V(t)$$ is the normal velocity of deformation, $$\delta$$ is the first variation, and $$\lambda_1,\;\lambda_2$$ are the Lagrange multipliers arising from the constraints. Let $$\langle \cdot,\cdot\rangle$$ be the inner product in $$L^2(\Sigma)$$. The Lagrange multipliers $$\lambda_1,\;\lambda_2$$ will be uniquely determined if and only if $G(\Sigma(t)) = \text{det} \begin{pmatrix} \left\langle\, \delta{\mathcal A}(\Sigma(t)),\, \delta{\mathcal A}(\Sigma(t)) \,\right\rangle & \left\langle\, \delta{\mathcal V}(\Sigma(t)),\, \delta{\mathcal A}(\Sigma(t)) \,\right\rangle \\ \left\langle\, \delta{\mathcal A}(\Sigma(t)),\, \delta{\mathcal V}(\Sigma(t)) \,\right\rangle & \left\langle\, \delta{\mathcal V}(\Sigma(t)),\, \delta{\mathcal V}(\Sigma(t)) \,\right\rangle \end{pmatrix}$ is non-zero.
The authors’ main results convert the Helfrich flow equation above into the form of a projected gradient flow. In case $$\Sigma_0$$ is sufficiently smooth and $$G(\Sigma_0)\neq 0$$, they use a method motivated by Y. Kohsaka and T. Nagasawa [Differ. Integral Equ. 19, No. 2, 121–142 (2006; Zbl 1212.49059)] to prove short time existence and uniqueness of a Helfrich flow with $$\Sigma(0) = \Sigma_0$$. They also obtain an appropriate result in case $$G(\Sigma_0) = 0$$ holds.

### MSC:

 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 49Q10 Optimization of shapes other than minimal surfaces 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35K30 Initial value problems for higher-order parabolic equations

### Keywords:

Helfrich variational problem; gradient flow; constraints

Zbl 1212.49059
Full Text: