arXiv subjects

Tristan Bullion-Gauthier

Publications and source records attributed to Tristan Bullion-Gauthier.

3 recordsLinked to original sources

arXiv2026

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

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 ${\mathscr E}$ has $L^2$ density. Our argument relies on a careful analysis of nonlocal variational inequalities, which may be of independent interest.

math.AP↗

arXiv2026

Existence of extremal functions in higher-order affine Sobolev inequalities

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces of arbitrary regularity, might be of independent interest.

math.FA↗

arXiv2025

Higher-order affine Sobolev inequalities

Zhang refined the classical Sobolev inequality $\|f\|_{L^{Np/(N-p)}} \lesssim \| \nabla f \|_{L^p}$, where $1\leq p \lt N$, by replacing $\|\nabla f\|_{L^p}$ with a smaller quantity invariant by unimodular affine transformations. The analogue result in homogeneous fractional Sobolev spaces $\mathring{W}^{s,p}$, with $0 \lt s \lt 1$ and $sp \lt N$, was obtained by Haddad and Ludwig. We generalize their results to the case where $s \gt 1$. Our approach, based on the existence of optimal unimodular transformations, allows us to obtain various affine inequalities, such as affine Sobolev inequalities, reverse affine inequalities, and affine Gagliardo-Nirenberg type inequalities. In a different but related direction, we also answer a question concerning reverse affine inequalities, raised by Haddad, Jiménez, and Montenegro.

math.FA↗