Discontinuous eikonal equations in metric measure spaces

Liu, Qing; Shanmugalingam, Nageswari; Zhou, Xiaodan

doi:10.1090/tran/9294

Discontinuous eikonal equations in metric measure spaces
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by Qing Liu, Nageswari Shanmugalingam and Xiaodan Zhou;

Trans. Amer. Math. Soc. 378 (2025), 695-729

DOI: https://doi.org/10.1090/tran/9294

Published electronically: September 25, 2024

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Abstract:

In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely $p$-integrable in the domain with a finite $p$. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Hölder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term.

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