Mathematics

The talk aims to discuss the existence and regularity problem for spacelike hypersurfaces $M$ in Lorentzian manifolds whose mean curvature is a prescribed, possibly singular distribution. When considering ambient Minkowski space, this leads to study the following Dirichlet problem for the function that (locally) describes $M$ as a graph, which we call $(\mathcal{BI})$: $$     \left\{ \begin{array}{ll}         -{\rm div} \left( \frac{Du}{\sqrt{1-|Du|^2}}\right) = \rho & \quad \text{in a domain } \, \Omega \Subset \R^n, \\[0.5cm]         u = \phi & \quad \text{on } \, \partial \Omega,        \end{array}     \right.$$for a given measure $\rho$ (the prescribed mean curvature) and boundary data $\phi$. Problem $(\mathcal{BI})$ also appears in Born-Infeld's theory of electrostatics, according to which $u$ describes the electric potential generated by the charge $\rho$. Even though $(\mathcal{BI})$ is formally the Euler-Lagrange equation of a nice convex functional $I_\rho$, the lack of smoothness of $I_\rho$ where $|Du|=1$ (i.e. where the graph of $u$ becomes lighlike) may prevent the unique variational minimizer $u_\rho$ to solve $(\mathcal{BI})$. As we shall see, this possibility actually occurs. We shall describe both existence and non-existence results helping to guess the possible sharp thresholds on $\rho$. A chief difficulty comes from the possible presence of ``light segments" in the graph of $u_\rho$, a fact that we will investigate in detail. Various open problems and research directions will be discussed.The talk is based on joint works with J. Byeon, N. Ikoma, A. Malchiodi and L. Maniscalco, available at arXiv:2112.11283 and arXiv:2512.17670.