Regarding the Euler–Plateau problem with elastic modulus

On p-Willmore Disks with Boundary Energies

“…[25]) based on the holomorphicity of the Hopf differential ( [21]) can be applied to conclude that a constant mean curvature (in particular, minimal) surface of disk type bounded by an asymptotic curve is necessarily a planar domain. For details, see Theorem 4.1 of [27], or also Theorem 4.1 of [16]. From this, it follows that K = H = 0 hold on Σ and κ n = τ g = 0 hold on ∂Σ.…”

“…θ ≡ ±π/2. When this happens, (16) implies that κ n ≡ 0 holds along the boundary of the surface, so that the boundary is represented by an asymptotic curve. In this setting, the shape of "perfectly wetted" equilibria of the total energy E[X], (8), is highly restricted.…”

“…In this section we will fix p = 0 so that the energy F [X], (32), reduces to an extended Kirchhoff-Plateau energy with elastic modulus (see [16] and references therein). In particular, the interior surface energy is the classical surface area.…”

“…Next, as done in Theorem 4.6, we adapt an argument due to Nitsche (see [25]) involving the Hopf differential (for details, see again [16] or [27]) to prove that a disk type constant mean curvature H ≡ H o = c o surface satisfying κ n ≡ 0 along the boundary is a planar domain. satisfying κ = 0.…”

Closed 1/2-Elasticae in the 2-Sphere

“…In the case of smooth spherical curves, the only closed critical trajectories are geodesics. Surprisingly, smooth closed spherical unconstrained p-elasticae other than geodesics are 2-elasticae or else p ∈ (0, 1) [20]. In [2,4,20,32,33], for 0 < p < 1, the existence of infinitely many closed unconstrained p-elasticae in S 2 was shown.…”