leon-bungert / Infinity-Laplacian-Eigenfunctions

This code can be used to compute eigenfunctions of the infinity Laplace operator on general domains.

Infinity Laplacian Eigenfunctions

This code produces the examples for the paper "The Infinity Laplacian eigenvalue problem: reformulation and a numerical scheme ": https://arxiv.org/abs/2004.08127

@misc{bozorgnia2020infinity,
    title={The infinity Laplacian eigenvalue problem: reformulation and a numerical scheme},
    author={Farid Bozorgnia and Leon Bungert and Daniel Tenbrinck},
    year={2020},
    eprint={2004.08127},
    archivePrefix={arXiv},
    primaryClass={math.NA}
}

It computes eigenfunctions of the infinity Laplacian, i.e., functions $u\in W^{1,\infty}_0(\Omega)$ which are viscosity solutions of the PDE

$$ \begin{align*} \begin{cases} \min(|\nabla u| - \lambda u, -\Delta_\infty u) = 0,\quad &u>0 ,\\ -\Delta_\infty u = 0,\quad &u=0 ,\\ \max(-|\nabla u| - \lambda u, -\Delta_\infty u) = 0,\quad &u<0. \end{cases} \end{align*} $$

The file RUN_ME_eigenfunction.m computes ground states, i.e., positive solutions, on different domains.

The file RUN_ME_second_eigenfunction.m computes second eigenfunctions on several symmetric domains. For the square, even higher eigenfunctions can be computed.

The file RUN_ME_comparison.m gives a numerical confirmation of the recently proved statement (https://arxiv.org/abs/2210.03447) that the infinity harmonic potential on a square is no ground state.