Given a measure ρ on a domain Ω ⊂ Rm , we study spacelike graphs over Ω in Minkowski space with Lorentzian mean curvature ρ and Dirichlet boundary condition on ∂Ω , which solve [Figure not available: see fulltext.] The graph function also represents the electric potential generated by a charge ρ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer uρ of the associated action Iρ(ψ)≐∫Ω(1-1-|Dψ|2)dx-⟨ρ,ψ⟩ among functions ψ satisfying | Dψ| ≤ 1 , by the lack of smoothness of the Lagrangian density for | Dψ| = 1 one cannot guarantee that uρ satisfies the Euler-Lagrange equation (BI). A chief difficulty comes from the possible presence of light segments in the graph of uρ . In this paper, we investigate the existence of a solution for general ρ . In particular, we give sufficient conditions to guarantee that uρ solves (BI) and enjoys log -improved energy and Wloc2,2 estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of ρ to ensure the solvability of (BI).
Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model / J. Byeon, N. Ikoma, A. Malchiodi, L. Mari. - In: ANNALS OF PDE. - ISSN 2199-2576. - 10:1(2024), pp. 4.1-4.86. [10.1007/s40818-023-00167-4]
Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model
L. MariUltimo
2024
Abstract
Given a measure ρ on a domain Ω ⊂ Rm , we study spacelike graphs over Ω in Minkowski space with Lorentzian mean curvature ρ and Dirichlet boundary condition on ∂Ω , which solve [Figure not available: see fulltext.] The graph function also represents the electric potential generated by a charge ρ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer uρ of the associated action Iρ(ψ)≐∫Ω(1-1-|Dψ|2)dx-⟨ρ,ψ⟩ among functions ψ satisfying | Dψ| ≤ 1 , by the lack of smoothness of the Lagrangian density for | Dψ| = 1 one cannot guarantee that uρ satisfies the Euler-Lagrange equation (BI). A chief difficulty comes from the possible presence of light segments in the graph of uρ . In this paper, we investigate the existence of a solution for general ρ . In particular, we give sufficient conditions to guarantee that uρ solves (BI) and enjoys log -improved energy and Wloc2,2 estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of ρ to ensure the solvability of (BI).File | Dimensione | Formato | |
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