Speaker
Description
In this talk we will discuss the spacelike capillary problem in a cylinder of Minkowski spacetime $\mathbb{L}^{m+1}$: given a bounded domain $\Omega\subseteq \R^m$, it asks whether there exists a spacelike graph in the cylinder over $\Omega$ with prescribed mean curvature and Lorentzian contact angle with the wall of the cylinder. From a PDE point of view, this is a quasi-linear elliptic problem with a non-linear oblique derivative boundary condition and arises as the Euler--Lagrange equation of an area-like functional. The ellipticity of the equation depends on the slope of the graph and degenerates as it approaches the light direction, making regularity issues quite subtle. A major difficulty is the possible presence of light segments in the graph of the maximizer of the functional. We will present existence and regularity results for this problem over convex domains of $\R^m$, when the prescribed mean curvature and angle are bounded and Lipschitz, respectively.