On Pleijel-type nodal domain bounds for the $p$-Laplacian

Abstract

We provide an upper estimate à la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{{\lambda }_k(p;\Omega )\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k\to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality ${\lambda }_2(p;B)\le \cdots \le {\lambda }_{N+1}(p;B)\le {\lambda }_{\ominus }(p)$ in an $N$-dimensional ball $B$, where ${\lambda }_{\ominus }(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.

18 pages, 1 figure