Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator

Recently, Haddad, Jiménez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator ${\mathrm{\Delta }}_{Q,p}^{\mathcal{A}}f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval. Given $n,m\in \mathbb{N}$, a sufficiently smooth convex body $Q\subset {\mathbb{R}}^m$, a bounded, open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and $p>1$, we investigate the eigenvalue problem $$\{\begin{array}{ll}{\mathrm{\Delta }}_{Q,p}^{\mathcal{A}}f={\lambda }_{1,p}^{\mathcal{A}}(Q,\mathrm{\Omega })\mathrm{\mid }f{\mathrm{\mid }}^{p-2}f& \text{ in }\mathrm{\Omega };\\ f=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ for $f\in W_0^{1,p}(\mathrm{\Omega })$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities.