$L^2$-regularity of minimizers of anisotropic interaction functionals

RESUMO

We consider interaction functionals of the form \begin{equation*} \nu\mapsto {\mathscr E}(\nu)=\int\_{{\mathbb R}^N}\int\_{{\mathbb R}^N} W(x-y)\intd\nu(y)\intd\nu(x)+\int\_{{\mathbb R}^N} V(x)\intd\nu(x)\text{,} \end{equation*} involving an anisotropic kernel $W$ and a general confinement potential $V$. Under standard assumptions on $W$ and its Fourier transform, we show that the minimizer of ${\mathscr E}$ has $L^2$ density. Our argument relies on a careful analysis of nonlocal variational inequalities, which may be of independent interest.