Supersymmetric factorization and Bogomolnyi bounds for Lp embeddings with p ≠ 4

Investigate whether Bogomolnyi’s bound and the associated supersymmetric factorization method extend to magnetically improved embeddings into Lp(ℝ^2) for 2 < p < ∞ in the self-generated field setting A[|u|^2], beyond the p = 4 case treated in this work.

Background

The main results rely crucially on a supersymmetric (Bogomolnyi) factorization identity tailored to the L⁴ framework, which underpins exact lower bounds and characterization of minimizers for β ≥ 2. The authors note that magnetically improved Sobolev embeddings can be formulated for general L^p targets (2 < p < ∞), and diamagnetic lower bounds remain valid in that setting.

However, it is unclear whether the Bogomolnyi bound and the supersymmetric factorization they use at p = 4 carry over to p ≠ 4 in the self-generated magnetic field framework. Establishing such an extension would generalize the core method and potentially yield analogous results for a wider class of interpolation inequalities.

References

As considered by Dolbeault et al. in the case of the external field (see also ), one may study magnetically improved embedding into any intermediate $L^p$ space, $2 < p < \infty$, and again obtain lower bounds by means of the diamagnetic inequality eq:diamag, however, it is a priori not clear whether Bogomolnyi's bound and our crucial supersymmetric factorization method will be applicable if $p \neq 4$.

eq:diamag:

$\int_{\R^2} |\nabla_{A}u|^2 \ge \int_{\R^2} \left|\nabla|u|\right|^2,$

— A generalized Liouville equation and magnetic stability (2404.09332 - Ataei et al., 14 Apr 2024) in Remark “Extended interpolation”, Section 1.2

Supersymmetric factorization and Bogomolnyi bounds for Lp embeddings with p ≠ 4

Background

References

Related Problems