$L^2$ estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator

TL;DR

This paper establishes $L^2$ bounds for eigenfunctions of the Tricomi operator on specific domains, revealing how these bounds depend on domain geometry and parameters, advancing understanding of eigenfunction behavior in mixed-type PDEs.

Contribution

It introduces a family of domains for the Tricomi operator and derives $L^2$ estimates for eigenfunctions, showing their dependence on domain parameters and shape.

Findings

Eigenfunction estimates depend on domain parameters $ta$ and $$

Domains are $D$-star-shaped iff $ta \u2265 1/2$

Results connect domain geometry with eigenfunction bounds

Abstract

We introduce a family ${\cal F}$ of normal Tricomi domains $\Om_{\a,\b}$, $\a>0>\b$, and we show that its elements are $D$-star-shaped with respect to the vector field $D=-3x\partial_x-2y\partial_y$ if and only if $\a\ge1/2$. Provided that the underlying domain $\Om$ belongs to ${\cal F}$ for some $\a\ge1/2$, we then establish $L^2$ estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator. In particular, our result highlights a dependency of these estimates on the values of $\a$ and $\b$ and the parabolic diameter of $\Om$.