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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

Citations:

Zbl 1212.49059
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