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

Abstract

We consider interaction functionals of the form \begin{equation*} ν\mapsto {\mathscr E}(ν)=\int\_{{\mathbb R}^N}\int\_{{\mathbb R}^N} W(x-y)\intdν(y)\intdν(x)+\int\_{{\mathbb R}^N} V(x)\intdν(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 $\mathcal{E}$ has $L^2$ density. Our argument relies on a careful analysis of nonlocal variational inequalities, which may be of independent interest.