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I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds. I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...