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Numerical Results – Optimal Shapes – Dirichlet Eigenvalues – Perimeter Constraint

Recently it was proved that the problem

$\displaystyle \min_{|\Omega|=1} \lambda_k(\Omega)$

has a solution even if ${\Omega}$ is not confined to a bounded open set ${D}$. (${\lambda_k(\Omega)}$ is the ${k}$-th eigenvalue for the Laplace operator with Dirichlet boundary conditions.)

A similar result was proven recently by B. Velichkov and G. de Philippis for the case where the volume constraint is replaced by a perimeter constraint, i.e. the problem

$\displaystyle \min_{\text{Per} (\Omega)=1}\lambda_k(\Omega)$

has a solution, and this solution is regular enough.

The existence result motivated the following numerical study of the first ${20}$ optimal eigenvalues in 2D. First note that the optimal shape for the first eigenvalue is a ball, as a consequence of the Faber-Krahn inequality and the isoperimetric inequality. The second eigenvalue was studied in this paper, where existence results and a numerical computed shape was presented.

For ${k=3}$ it is conjectured that the optimal shape for ${\lambda_3(\Omega)}$ for the volume constraint is the ball. This fact, coupled with the isoperimetric inequality imply that if the mentioned conjecture is true, then a ball minimizes ${\lambda_3(\Omega)}$ under perimeter constraint. This is confirmed in the numerical results below.

Another particularity of the 2D case is that optimal shapes must always be convex, because convexification decreases the perimeter and the eigenvalue at the same time. Convex shapes are in particular star-convex, which allows us to characterize the boundary using polar coordinates, and then use the same method described in this post.

The algorithm minimizes $\lambda_k(\Omega)+\text{Per}(\Omega)$. It can be proved that the optimal shapes of the initial problem and the optimal shapes of this problem are homothetic.

One of the interesting results of this simulation was finding values of $k$ for which the eigenvalue is not multiple at the optimum. This differs from the case of the volume constraint, where it is conjectured that the eigenvalues are always multiple at the optimum, and the multiplicity is increasing as $k$ increases.

lambda_1 = 11.55

lambda_2 = 15.28 (simple)

lambda_3 = 15.75 (double)

lambda_4 = 18.35 (double)

lambda_5 = 19.11 (double)

lambda_6 = 20.09 (simple)

lambda_7 = 21.50 (double)

lambda_8 = 22.02 (double)

lambda_9 = 23.20 (simple)

lambda_10 = 23.55 (double)

lambda_11 = 24.60 (double)

lambda_12 = 24.74 (triple)

lambda_13 = 25.98 (simple)

lambda_14 = 26.44 (double)

lambda_15 = 26.91 (simple)

lambda_16 = 27.26 (double)

lambda_17 = 27.36 (double)

lambda_18 = 28.63 (double)

lambda_19 = 29.08 (double)

lambda_20 = 29.52 (triple)

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