Double covering with cutting on the red line

Rappelons que les valeurs propres sur le carré sont donnés par λ_i,j=π²(i²+j²) (=19.7392, 49.3480, 49.3480, 78.9568, 98.6960, 98.6960, 128.3049, 128.3049, 167.7833, 167.7833, ...)

On remarque que pour toutes les configurations, le 6e vecteur propre ne donne jamais de 3-partition sauf lorsque la coupure part du centre du carré. Par ailleurs, la 6e valeur propre est toujours strictement inférieure à celle obtenue lorsque la coupure part du centre du carré.
En ce qui concerne la 7e valeur propre, elle est minimale lorsque la coupure part du centre du carré. Le 7e vecteur prore ne donne pas toujours une 3-partition. Notons que λ₇≈8π² lorsque la coupure est petite. Quand λ₇ < 8π², nous avons alors λ₈≈8π² (la 4e valeur propre du carré).

On constate que quelque soit le point intérieur choisi pour effectuer le recouvrement, les valeurs propores λ₁, λ₄ et λ₅ restent constantes, égales respectivement à 19.7392, 49.3480 et 49.3480, qui sont les premières valeurs propres du Laplacien-Dirichlet sur le carré d'aire 1. Les figures suivantes donnent le graphe de λ₂, λ₃, λ₆, λ₇ et λ₈ en fonction de la position du centre de coupure. On voit clairement que λ₆ est maximale lorsque la coupure part du centre du carré alors que λ₇ est minimale dans ce cas. On retrouve également les symétries du carré.
Le graphe des 7e et 8e valeurs propres présente des paliers qui viennent du fait que ces valeurs propres sont, soit l'un, soit l'autre, des valeurs propres de la base quand le vecteur propre associé est antisymétrique sur le revêtement. On remarque aussi que λ₂ est proche de 2π² lorsque le recouvrement se fait le long d'un segment court et tend vers la valeur 33.513786 lorsque le point singulier tend vers le centre du carré. En ce qui concerne λ₃, la valeur propre est proche de 5π² lorsque le recouvrement se fait le long d'un segment court et tend également vers la valeur 33.513786 lorsque le point singulier tend vers le centre du carré.
Ainsi, lorsque le segment de recouvrement est petit, on retrouve les valeurs propres du carré.

Le tableau suivant donne les résultats pour diverses configurations sur le carré [0,1]x[0,1] : la ie ligne et je colonne correspond à la configuration où la coupure part du sommet de coordonnées (0,0) et le centre a pour coordonées (x_i,y_j)=1/10+3(i,j)/20, avec 0≤i≤6 et 0≤j≤6 .
Chacune des cases du tableau donne le domaine avec le lieu de la coupure en première ligne, les 8 premières valeurs propres en deuxième ligne, le signe des 8 premiers vecteurs propres sur le premier feuillet en troisième ligne.
Les lignes nodales des 12 premiers vecteurs propres sont visibles sur cette page.
L'évolution des lignes nodales lorsque le point de coupure bouge le long de la diagonale ou de la médiatrice est représentée sur cette page.

&lambda1 = 19.739209 &lambda2 = 19.743060 &lambda3 = 49.348021 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.377699 &lambda7 = 78.956835 &lambda8 = 79.013922	&lambda1 = 19.739209 &lambda2 = 19.761789 &lambda3 = 49.348001 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.509210 &lambda7 = 78.956835 &lambda8 = 79.242364	&lambda1 = 19.739209 &lambda2 = 19.792378 &lambda3 = 49.347889 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.685939 &lambda7 = 78.956835 &lambda8 = 79.471825	&lambda1 = 19.739209 &lambda2 = 19.828129 &lambda3 = 49.347588 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.835065 &lambda7 = 78.956835 &lambda8 = 79.514923	&lambda1 = 19.739209 &lambda2 = 19.861807 &lambda3 = 49.347060 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.919556 &lambda7 = 78.956835 &lambda8 = 79.329461	&lambda1 = 19.739209 &lambda2 = 19.885219 &lambda3 = 49.346457 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.944038 &lambda7 = 78.956835 &lambda8 = 79.067799	&lambda1 = 19.739209 &lambda2 = 19.893132 &lambda3 = 49.346181 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.944061 &lambda7 = 78.949513 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.761779 &lambda3 = 49.348001 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.509140 &lambda7 = 78.956835 &lambda8 = 79.242245	&lambda1 = 19.739209 &lambda2 = 19.869754 &lambda3 = 49.347196 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 50.190911 &lambda7 = 78.956835 &lambda8 = 80.313290	&lambda1 = 19.739209 &lambda2 = 20.043803 &lambda3 = 49.342824 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 51.091520 &lambda7 = 78.956835 &lambda8 = 81.419268	&lambda1 = 19.739209 &lambda2 = 20.249203 &lambda3 = 49.330757 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 51.889437 &lambda7 = 78.956835 &lambda8 = 81.707272	&lambda1 = 19.739209 &lambda2 = 20.445644 &lambda3 = 49.308866 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.379046 &lambda7 = 78.956835 &lambda8 = 80.706344	&lambda1 = 19.739209 &lambda2 = 20.588115 &lambda3 = 49.283308 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.553358 &lambda7 = 78.956835 &lambda8 = 79.278308	&lambda1 = 19.739209 &lambda2 = 20.640456 &lambda3 = 49.271196 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.577517 &lambda7 = 78.661342 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.792379 &lambda3 = 49.347890 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.685944 &lambda7 = 78.956835 &lambda8 = 79.471825	&lambda1 = 19.739209 &lambda2 = 20.043746 &lambda3 = 49.342818 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 51.091282 &lambda7 = 78.956835 &lambda8 = 81.419106	&lambda1 = 19.739209 &lambda2 = 20.448209 &lambda3 = 49.314564 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.959912 &lambda7 = 78.956835 &lambda8 = 83.722908	&lambda1 = 19.739209 &lambda2 = 20.938295 &lambda3 = 49.234151 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 54.722366 &lambda7 = 78.956835 &lambda8 = 84.823892	&lambda1 = 19.739209 &lambda2 = 21.424731 &lambda3 = 49.080375 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 55.925175 &lambda7 = 78.956835 &lambda8 = 82.224675	&lambda1 = 19.739209 &lambda2 = 21.793182 &lambda3 = 48.892024 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 56.429686 &lambda7 = 78.479683 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 21.933237 &lambda3 = 48.799694 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 56.521462 &lambda7 = 77.037408 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.828046 &lambda3 = 49.347588 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.834633 &lambda7 = 78.956835 &lambda8 = 79.514463	&lambda1 = 19.739209 &lambda2 = 20.249726 &lambda3 = 49.330764 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 51.891755 &lambda7 = 78.956835 &lambda8 = 81.709605	&lambda1 = 19.739209 &lambda2 = 20.937820 &lambda3 = 49.233924 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 54.720665 &lambda7 = 78.956835 &lambda8 = 84.821723	&lambda1 = 19.739209 &lambda2 = 21.804610 &lambda3 = 48.942410 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 57.551307 &lambda7 = 78.956835 &lambda8 = 88.547689	&lambda1 = 19.739209 &lambda2 = 22.721534 &lambda3 = 48.346951 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 59.675465 &lambda7 = 78.956835 &lambda8 = 81.662865	&lambda1 = 19.739209 &lambda2 = 23.470106 &lambda3 = 47.572155 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 60.710940 &lambda7 = 75.083453 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 23.767212 &lambda3 = 47.181324 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 60.929869 &lambda7 = 73.065033 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.861834 &lambda3 = 49.347059 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.919670 &lambda7 = 78.956835 &lambda8 = 79.329495	&lambda1 = 19.739209 &lambda2 = 20.445806 &lambda3 = 49.308810 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.379514 &lambda7 = 78.956835 &lambda8 = 80.705701	&lambda1 = 19.739209 &lambda2 = 21.425157 &lambda3 = 49.080397 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 55.926782 &lambda7 = 78.956835 &lambda8 = 82.227270	&lambda1 = 19.739209 &lambda2 = 22.720164 &lambda3 = 48.344940 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 59.672097 &lambda7 = 78.956835 &lambda8 = 81.647373	&lambda1 = 19.739209 &lambda2 = 24.216847 &lambda3 = 46.743931 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 62.639312 &lambda7 = 75.340055 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 25.601941 &lambda3 = 44.614298 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 64.192621 &lambda7 = 70.269565 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 26.221910 &lambda3 = 43.547710 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 64.567057 &lambda7 = 68.802507 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.885119 &lambda3 = 49.346457 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.943619 &lambda7 = 78.956835 &lambda8 = 79.067647	&lambda1 = 19.739209 &lambda2 = 20.587993 &lambda3 = 49.283352 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.553103 &lambda7 = 78.956835 &lambda8 = 79.279120	&lambda1 = 19.739209 &lambda2 = 21.793005 &lambda3 = 48.892304 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 56.429101 &lambda7 = 78.480366 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 23.470115 &lambda3 = 47.575241 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 60.710748 &lambda7 = 75.091096 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 25.601868 &lambda3 = 44.614081 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 64.192697 &lambda7 = 70.269622 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 28.000859 &lambda3 = 40.646085 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 65.937154 &lambda7 = 67.521978 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 29.420617 &lambda3 = 38.464350 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 66.313233 &lambda7 = 66.863005 &lambda8 = 78.956835
&lambda1 = 19.739209 &lambda2 = 19.893749 &lambda3 = 49.346182 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 49.946395 &lambda7 = 78.949519 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 20.640835 &lambda3 = 49.271177 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 52.578780 &lambda7 = 78.661272 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 21.932724 &lambda3 = 48.799821 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 56.520008 &lambda7 = 77.037816 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 23.767718 &lambda3 = 47.182878 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 60.930871 &lambda7 = 73.067945 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 26.221200 &lambda3 = 43.547413 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 64.566363 &lambda7 = 68.802277 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 29.423328 &lambda3 = 38.468137 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 66.313831 &lambda7 = 66.863617 &lambda8 = 78.956835	&lambda1 = 19.739209 &lambda2 = 33.513786 &lambda3 = 33.518037 &lambda4 = 49.348022 &lambda5 = 49.348022 &lambda6 = 66.581166 &lambda7 = 66.581166 &lambda8 = 78.956835