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arXiv · 2608.00887

Free boundary space-like graphs with prescribed mean curvature

Lorenzo Maniscalco
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Abstract

We address the prescribed Lorentzian mean curvature problem over a convex bounded domain $Ω$ of $\mathbb R^m$ with bounded right-hand side and homogeneous capillary boundary condition. We prove that the problem has a unique $W^{2,2}$-regular weak solution $u$ with zero mean and that $|Du| \leq 1 - θ$ for some $θ\in(0,1)$ only depending on the data. Such $u$ is also the unique maximizer of an associated functional. A key step in proving that the maximizer is a weak solution consists in showing that it has no light segments, i.e. segments along which $|Du| = 1$. This holds for arbitrary bounded capillary boundary data and can thus be an interesting result on its own.

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Lorenzo Maniscalco. 2026-08-01. Free boundary space-like graphs with prescribed mean curvature. https://arxiv.org/abs/2608.00887

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