We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.

We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.

The phenomenon of dispersion in a physical system occurs whenever the elementary building blocks of the system, whether they are particles or waves, overall move away from each other, because each evolves according to a distinct momentum. This physical process limits the superposition of particles or waves, and leads to remarkable mathematical properties of the densities or amplitudes, including local and global decay, Strichartz estimates, and smoothing.

In kinetic theory, the effects of dispersion in the whole space were notably well captured by the estimates developed by Castella and Perthame in 1996, which, for instance, are particularly useful in the analysis of the Boltzmann equation to construct global solutions. However, these estimates are based on the transfer of integrability of particle densities in mixed Lebesgue spaces, which fails to apply to general settings of kinetic dynamics.

Therefore, we are now interested in characterizing the kinetic dispersive effects in the whole space in cases where only natural principles of conservation of mass, momentum and energy, and decay of entropy seem to hold. Such general settings correspond to degenerate endpoint cases of the Castella–Perthame estimates where no dispersion is effectively measured. However, by introducing a suitable kinetic uncertainty principle, we will see how it is possible to extract some amount of entropic dispersion and, in essence, measure how particles tend to move away from each other, at least when they are not restricted by a spatial boundary.

A simple application of entropic dispersion will then show us how kinetic dynamics in the whole space inevitably leads, in infinite time, to an asymptotic thermodynamic equilibrium state with no particle interaction and no available heat to sustain thermodynamic processes, thereby providing a provocative interpretation of the heat death of the universe.

We will consider a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and (sub)critical nonlinearities. In the critical regime, we show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. Moreover, through a suitable nonlinear eigenvalue problem, we prove convergence of solutions in dependence of the exponents of the nonlinearities in the (sub)critical range. Finally, I will briefly discuss related results on multiplicity, symmetry breaking, and regularity.

Based on joint works with A. Pistoia, A. Saldaña and H. Tavares.

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

\[ \min\Big\{\min_{u\in H^1_0(\Omega),\;\int u^2=1}\Big\{\int_\Omega|\nabla u|^2+q \int_\Omega\int_\Omega\frac{u^2(x)u^2(y)}{|x-y|}\,dx\,dy\Big\} : \Omega\subset \mathbb{R}^3,\;|\Omega|=1\Big\}, \]

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).