Mirror flows and $p$-Laplacian eigenvalue problems on metric measure spaces

We study mirror flows as Banach-space counterparts of Hilbertian gradient flows and show that they play an essential role in $p$-Laplacian eigenvalue problems on metric measure spaces. Under a Rellich–Kondrachov type compactness assumption and for $p\ge 2$, we establish a Ljusternik–Schnirelman type existence theorem without assuming $C^1$-regularity of the associated energy. More precisely, we prove that every element $\lambda$ of the Krasnoselskii spectrum is an eigenvalue of the $p$-Laplacian ${\mathrm{\Delta }}_p$; that is, there exists a nontrivial solution $f$ to ${\mathrm{\Delta }}_{p}f=-\lambda \mathrm{\mid }f{\mathrm{\mid }}^{p-2}f$.We also investigate the large time behavior of the corresponding mirror flow and prove its convergence to an eigenfunction when the eigenvalue is simple and isolated.