Some existence results for the optimisation of eigenvalues of biharmonic operators
Venue: Idriss Mazari - Luogo: Aula Buzano (DISMA Politecnico di Torino)
In this talk, I will present the results of a recent collaboration with E. Davoli and U. Stefanelli. More precisely, we investigate the existence and characterisation of optimal designs for the spectral optimisation problem: minimise the first eigenvalue of $\Delta (D(x) \Delta)-g(x)$ with respect to the density $D$, under pointwise and $L^1$ constraints. It is well-known that such problems can exhibit homogenisation phenomena and, consequently, that existence might not hold, as is the case in the standard two-phase problems. Here, we will prove the existence of optimal profiles $D^*$, drawing some inspiration and comparison with the literature devoted to two-phase problems. We will highlight the similarities and differences in these two settings.
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