# Connected surfaces with boundary minimizing the Willmore energy

@article{Novaga2019ConnectedSW, title={Connected surfaces with boundary minimizing the Willmore energy}, author={Matteo Novaga and Marco Pozzetta}, journal={Mathematics in Engineering}, year={2019} }

For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $\gamma=\gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $<4\pi$, there exists a…

## 10 Citations

### On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 𝔤 ≥ 1 having the curve Γ as boundary, without any…

### Boundary value problems for the Helfrich functional for surfaces of revolution

- Mathematics
- 2020

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor $\varepsilon\ge 0$. We…

### Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour

- MathematicsCalculus of Variations and Partial Differential Equations
- 2021

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor $$\varepsilon \ge 0$$ ε…

### Stationary surfaces with boundaries

- MathematicsAnnals of Global Analysis and Geometry
- 2022

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized “bending energy” functional…

### On the convergence of the Willmore flow with Dirichlet boundary conditions

- Mathematics
- 2023

Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and…

### A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds

- Mathematics
- 2022

For varifolds whose ﬁrst variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability,…

### Existence of varifold minimizers for the multiphase Canham–Helfrich functional

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature…

### Existence of varifold minimizers for the multiphase Canham–Helfrich functional

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature…

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In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of…

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We provide two sharp sufficient conditions for immersed Willmore surfaces in $$\mathbb{R }^3$$R3 to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These…

### On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 𝔤 ≥ 1 having the curve Γ as boundary, without any…

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A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝm is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising…

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Let $$\Omega $$Ω be a bounded domain in $$\mathbb {R}^n$$Rn, $$n\ge 3$$n≥3 with smooth boundary $$\partial \Omega $$∂Ω and a small hole. We give the first example of sign-changing bubbling solutions…

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We provide two sharp sufﬁcient conditions for immersed Willmore surfaces in R 3 to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out…

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Abstract We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct…