Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model. (English) Zbl 1539.35024
In this article, the authors show existence and regularity for prescribed Lorentzian mean curvature hypersurfaces and the Born-Infeld model given by
\[
\begin{cases}
\begin{aligned}
-\operatorname{div}\left(\frac{Du}{\sqrt{1-|Du|^{2}}}\right) =\rho \quad & \text{on } \Omega \subset \mathbb{R}^{m},\\
u=\phi \quad & \text{ on } \partial\Omega,
\end{aligned}
\end{cases}
\]
where \(\phi \in C(\partial \Omega)\), and \(D\) and \(|\cdot|\) are the connection (gradient) and norm in \(\mathbb{R}^{N}\). Moreover, the reader will be able to review known results in a bounded domain and in \(\mathbb{R}^{N}\) on this topic and the contributions of the authors in both cases.
Reviewer: Giovany Malcher Figueiredo (Brasília)
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35J62 | Quasilinear elliptic equations |
| 35R06 | PDEs with measure |
| 49J40 | Variational inequalities |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
Keywords:
prescribed Lorentzian mean curvature; Born-Infeld model; Euler-Lagrange equation; regularity of solutions; measure dataReferences:
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