Plateau-rayleigh instability of singular minimal surfaces

Rafael López

doi:10.3934/cpaa.2022086

Article Contents

2022, Volume 21, Issue 9: 2981-2997. Doi: 10.3934/cpaa.2022086

This issue Previous Article Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators Next Article Non-convex sweeping processes in the space of regulated functions

Plateau-rayleigh instability of singular minimal surfaces

Rafael López

Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

Received: July 31, 2021

Revised: March 31, 2022

Early access: May 2022

Published: September 2022

This work has been partially supported by the Projects I+D+i PID2020-117868GB-I00, A-FQM-139-UGR18 and P18-FR-4049

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Abstract

We prove a Plateau-Rayleigh criterion of instability for singular minimal surfaces, providing explicit bounds on the amplitude and length of the surface. More generally, we study the stability of $ \alpha $-singular minimal hypersurfaces considered as hypersurfaces in weighted manifolds. If $ \alpha<0 $ and the hypersurface is a graph, then we prove that the hypersurface is stable. If $ \alpha>0 $ and the surface is cylindrical, we give numerical evidences of the instability of long cylindrical $ \alpha $-singular minimal surfaces.

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Mathematics Subject Classification: Primary: 53C42, 53A10; Secondary: 58E30, 35J60, 35P30, 65L15.

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Figure 1. Left: corridor in the Colegio de las Teresianas, Barcelona ([25]). Right: the singular minimal surface $ \{(s,t,\cosh(s)):s,t\in{\mathbb R}\} $ constructed by repeating a catenary (blue) in a horizontal direction

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Figure 2. Solutions of (2.3) for values $ \alpha = -2 $ (right), $ \alpha = 0.8 $ (middle) and $ \alpha = 2 $ (right)

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Figure 3. Case $ \alpha = 1 $. Left: the function $ I_1(a) $. Right: the function $ L\mapsto I_2(a,L) $ (here $ a = 1 $)

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Figure 4. The function $ L_0 = L_0(a) $ given in (4.6)

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Table 1. Values of $ I(a,L) $ for $ \alpha = 2 $

$ L $	$ 1 $	$ 10 $	$ 20 $	$ 30 $	$ 40 $	$ 50 $
$ a $
0.2	-0.02389	-0.02031	-0.02028	-0.02028	-0.02028	-0.02028
0.3	-0.09371	-0.06398	-0.06375	-0.06371	-0.06370	-0.06369
0.4	-0.27824	-0.13500	-0.13392	-0.13372	-0.13365	-0.13362
0.5	-0.74222	-0.21609	-0.21210	-0.21137	-0.21111	-0.21099
0.6	-1.93468	-0.253826	-0.24109	-0.23873	-0.23790	-0.237527
0.7	-5.20226	-0.12268	-0.08420	-0.07708	-0.07458	-0.07343
0.8	-15.15280	$\fbox{0.45027} $	0.5684	0.5903	0.59802	0.60157
0.9	-50.9705	$ \fbox{2.0161}$	2.41751	2.49185	2.51786	2.52991

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Table 2. Case $ \alpha = 2 $. Values of $ I(0.72,L) $

$ L $	$ 22 $	$ 24 $	$ 26 $	$ 28 $	$ 30 $	$ 32 $
	-0.0053	-0.0032	-0.0015	-0.0002	$ \fbox{0.0007}$	0.0016

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Table 3. Case $ \alpha = 2 $. Values of $ I(a,L) $ when $ a>0.72 $

$ L $	$ 2 $	$ 3 $	$ 4 $	$ 5 $	$ 6 $	7	8	9
$ a $
0.75	-2.061	-0.820	-0.386	-0.185	-0.076	-0.010	$\fbox{0.032} $	0.061
0.80	-3.332	-1.143	-0.377	-0.022	$\fbox{0.170} $	0.286	0.361	0.413
0.90	-10.829	-3.395	-0.793	$ \fbox{0.410}$	1.064	1.459	1.715	1.890
1.00	-49.743	-17.942	-6.812	-1.660	$\fbox{1.137} $	2.824	3.920	4.670
1.05	-126.723	-49.655	-22.681	-10.196	-3.414	$ \fbox{0.674}$	3.328	5.148

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Table 4. Case $ \alpha = 3 $. Values of $ I(a,L) $ when $ a>0.53 $

$ L $	$ 4 $	$ 5 $	$ 6 $	$ 7 $	$ 8 $	$ 9 $	10	11	12
$ a $
0.54	-0.394	-0.216	-0.119	-0.060	-0.022	$ \fbox{0.003}$	0.021	0.035	0.046
0.56	-0.531	-0.247	-0.093	$\fbox{0.000} $	0.059	0.100	0.130	0.152	0.168
0.58	-0.906	-0.433	-0.176	-0.021	$\fbox{0.079} $	0.148	0.197	0.234	0.262
0.60	-1.928	-1.079	-0.618	-0.340	-0.160	-0.036	$ \fbox{0.051}$	0.117	0.167

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