Level curves of minimal graphs

We consider minimal graphs $u = u(x, y) \gt 0$ over domains $D \subset R^2$ bounded by an unbounded Jordan arc $\gamma$ on which $u = 0$.We prove an inequality on the curvature of the level curves of $u$, and prove that if $D$ is concave, then the sets $u(x, y) \gt C (C \gt 0)$ are all concave. A consequence of this is that solutions, in the case where $D$ is concave, are also superharmonic.

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Received 28 April 2019

Accepted 16 September 2019

Published 17 March 2023