Numerical Results – Optimal Shapes – Dirichlet Eigenvalues – Volume Constraint
The only known exact shapes which minimize the eigenvalues of the Laplace operator with Dirichlet condition are those for and . Nothing is proved for higher eigenvalues, but there are some numerical tests which were performed to find what the optimal shapes look like. Such tests were made first by Edouard Oudet for and recently by P. Freitas, P. Antunes for .
One way to do this is by representing the boundary of the set by its radial function. This is, of course, possible only if the domain is starshaped with respect to the origin (which is a reasonable assumption in this case, since the results of Oudet and Freitas, Antunes are starshaped). Moreover, in the following, we restrict to domains with smooth boundary, because of the possibility of using Fourier coefficients to describe a function on the interval . This is a very powerful tool, since a handful of parameters (20-40) can fully describe these optimal shapes as a consequence of the fact that the Fourier coefficients quickly decay to zero as .
After a talk with Braxton Osting and after reading one of his recent articles, I found out about the mpspack, a Matlab toolbox which solves efficiently the eigenvalue problem given the analytical expression of the boundary.
Another ingredient is an LBFGS implementation package in Matlab. LBFGS is a quasi-Newton optimization method and it has the advantage of being very fast when the number of variables is very large. To run the algorithm it is necessary to give it a function which calculates the value and the gradient of the functional we need to minimize. In our case, the variables will be the Fourier coefficients, the value of the function will be (calculated by the mpspack), and the gradient of this function will be calculated with the respect to the Fourier coefficients.
(Note that minimizing is equivalent to minimizing with due to the scaling of the eigenvalue and the area under homotheties)
I will present below the results I obtained using the method described above for . The results are comparable to the ones obtained by P. Freitas and P. Antunes with respect to the optimal shapes obtained and with respect to the values obtained. I will not present the cases since for these cases the optimal shapes are apparently not star convex, and this method cannot apply to them.