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Researcher: Yury Korolev, and Carola-Bibiane Schönlieb

We study minimisers of Rayleigh quotients involving $L^\infty$ type norms such as the $W^{1,\infty}$ Sobolev norm. Notable minimisers include ground states of the $\infty$-Laplacian $\Delta_\infty$, $\infty$-harmonic functions and the distance function.  We use convex analysis to characterise these minimisers and their subgradients. 

In certain settings, the distance function turns out to be the only global minimiser and the only _positive_ local minimiser, which yields an efficient gradient flow type algorithm for computing distance functions on graphs.

Related Publications 

Eigenvalue Problems in $\mathrm{L}^\infty$: Optimality Conditions, Duality, and Relations with Optimal Transport
L Bungert, Y Korolev – Comm. Amer. Math. Soc. 2 (2022), 345-373 (2021)