My curent work

My work is about shape optimization, it is about solving problems of the form

\[ \min\big\{F(\Omega)\,\big|\, \Omega\in\mathcal{A}\big\}, \]

where \(\mathcal{A} \subset \mathcal{P}(\mathbf{R}^d)\) represents the class of admissible shapes, typically open subsets of \(\mathbf{R}^d\) with given volume or subsets of a box \(D\subset\mathbf{R}^d\) and \(F\) is a given functional.

One of the main difficulty of the field is that in the general case there is no existence result for these problems particularly when the functional \(F\) is not monotonous for the inclusion of sets, and part of my research consists in studying the existence or not of solutions for problems involving such non monotonous functionals.

At the moment I specifically work on a problem related to spectral optimization for the Dirichlet \(p\)-Laplacian operator.

First, define the \(p\)-Laplacian first eigenvalue of an open set \(\Omega\),

\[\lambda_p(\Omega) = \inf_{u\in W^{1,p}_0(\Omega)} \frac{\int_\Omega |\nabla u|^p}{\int_\Omega |u|^p}, \]

I’m interested in the optimization of the non monotonous functional \(\mathcal{F}_{p,q}\) defined for \(q\leq p\) by

\[\mathcal{F}_{p,q}(\Omega) = \frac{\lambda_p(\Omega)^{1/p}}{\lambda_q(\Omega)^{1/q}}.\]

We are able to prove that \(\mathcal{F}_{p,q}\) admits both a minimizer and a maximizer among the open subsets of \(\mathbf{R}^d\) and we found some properties of the optimizers depending on the values of \(p\) and \(q\).