Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866909100771639296 |
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| author | Byeon, Jaeyoung Ikoma, Norihisa Malchiodi, Andrea Mari, Luciano |
| author_facet | Byeon, Jaeyoung Ikoma, Norihisa Malchiodi, Andrea Mari, Luciano |
| contents | Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action
$$
I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle
$$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11283 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model Byeon, Jaeyoung Ikoma, Norihisa Malchiodi, Andrea Mari, Luciano Analysis of PDEs Differential Geometry Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action $$ I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle $$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail. |
| title | Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2112.11283 |