Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many p-Laplacians
Abstract
We study the Dirichlet problem for an infinite series of p-Laplacians, −∑p=2∞apΔpu=f in Ω, u=g on ∂Ω, where {ap} is a sequence of nonnegative numbers whose power series has radius of convergence σ∈(0,∞]. The operator is formally −div( A(|∇ u|)∇ u) with $ {A}$ singular at |∇ u|=σ; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint |∇ u|∞≤σ, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux Λ:=∑p≥2apσp−1 as the quantity governing solvability of the equation: if Λ<∞ a weak solution exists only if |∫E f|≤ΛP(E) for every set of finite perimeter ESubsetΩ, so for f≡λ no solution exists once λ>Λh(Ω), h(Ω) the Cheeger constant of Ω. The threshold is sharp on balls, where the minimizer has a saturation region {|∇ u|=σ} of positive measure. When Λ=∞, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below σ is available; for f≡0 we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.
Related
- Determinants of Dirichlet and Neumann Laplacians on Constant Curvature Surfaces with Geodesic Polygonal Boundary
- Combinatorial Calabi flows on infinitely triangulated surfaces
- Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces
- Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature
- Gradient-Enhanced Proximal Algorithms for Mean Field Planning on Surfaces
BibTeX
@misc{hamid2026gradient,
title = {{Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many $p$-Laplacians}},
author = {Haydar Abdel Hamid and Juan Carlos Chata Ortiz and Francesco Petitta and Julio Daniel Rossi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28288},
eprint = {2609.28288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28288},
}
RIS
TY - UNPB
TI - Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many p-Laplacians
AU - Haydar Abdel Hamid
AU - Juan Carlos Chata Ortiz
AU - Francesco Petitta
AU - Julio Daniel Rossi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.28288
UR - https://arxiv.org/abs/2609.28288
AB - We study the Dirichlet problem for an infinite series of p-Laplacians, −∑_p=2^∞a_pΔ_pu=f in Ω, u=g on ∂Ω, where {a_p} is a sequence of nonnegative numbers whose power series has radius of convergence σ∈(0,∞]. The operator is formally −div( A(|∇ u|)∇ u) with $ {A}$ singular at |∇ u|=σ; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint |∇ u|_∞≤σ, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux Λ:=∑_p≥2a_pσ^p−1 as the quantity governing solvability of the equation: if Λ<∞ a weak solution exists only if |∫_E f|≤ΛP(E) for every set of finite perimeter ESubsetΩ, so for f≡λ no solution exists once λ>Λh(Ω), h(Ω) the Cheeger constant of Ω. The threshold is sharp on balls, where the minimizer has a saturation region {|∇ u|=σ} of positive measure. When Λ=∞, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below σ is available; for f≡0 we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.
ER -