In the last few years the Gamow problem, namely
\[ \min\Big\{P(\Omega)+\varepsilon \int_{\Omega}\int_\Omega \frac{1}{|x-y|}\,dx\,dy : \Omega\subset \mathbb{R}^3, |\Omega|=1\Big\}, \]
for $\varepsilon>0$, has attracted a lot of attention from mathematicians. Nowadays it is well understood that for small $\varepsilon$ there exist a minimizer and it is a ball, while for very large $\varepsilon$ there is no minimizer.
Although it is very easy to formulate, there are still several open problems about it (mostly concerning nonexistence of minimizers for large $\varepsilon$ in a generalized $N$-dimensional setting).
A variation of this model, which could be called ``spectral Gamow problem'' consists in using the first eigenvalue of the Dirichlet Laplacian instead of the Perimeter, namely to consider
\[ \min\Big\{\lambda_1(\Omega)+\varepsilon \int_{\Omega}\int_\Omega \frac{1}{|x-y|}\,dx\,dy : \Omega\subset \mathbb{R}^3, |\Omega|=1\Big\}, \]
and in this talk we will provide some new results on this case.
Moreover, we will consider a different problem but with a similar structure, which can be seen as the minimization of a Hartree functional settled in a box, namely
\[ \min\Big\{\min_{u\in H^1_0(\Omega),\;\int u^2=1}\Big\{\int_\Omega|\nabla u|^2+q \int_\Omega\int_\Omega\frac{u^2(x)u^2(y)}{|x-y|}\,dxdy\Big\} : \Omega\subset \mathbb{R}^3,\;|\Omega|=1\Big\}, \]
for $q>0$.
The study of this functional arises when describing the ground state of a superconducting charge qubit.
We show that there is a threshold $q_1>0$ such that for all $q\leq q_1$ existence of minimizers occurs and minimizers are $C^{2,\gamma}$ nearly spherical.
We will also give some ideas (although nonconclusive) on how to treat the nonexistence issue for this functional.
The techniques and tools needed in the proofs are very broad. We employ spectral quantitative inequalities, the regularity of free boundaries, spectral surgery arguments and shape variations.
This is a joint project with Cyrill Muratov (Pisa) and Berardo Ruffini (Bologna).
