Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many $p$-Laplacians
We study the Dirichlet problem for an infinite series of $p$-Laplacians, $$-\sum_{p=2}^{\infty}a_{p}Δ_{p}u=f \ \text{ in } Ω, \qquad u=g \ \text{ on } \partialΩ,$$ where $\{a_{p}\}$ is a sequence of nonnegative numbers whose power series has radius of convergence $σ\in(0,\infty]$. The operator is formally $-\operatorname{div}( {A}(|\nabla u|)\nabla u)$ with $ {A}$ singular at $|\nabla u|=σ$; 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 $\|\nabla u\|_\infty\leσ$, 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 $Λ:=\sum_{p\ge2}a_pσ^{p-1}$ as the quantity governing solvability of the equation: if $Λ<\infty$ a weak solution exists only if $|\int_E f|\leΛP(E)$ for every set of finite perimeter $E\SubsetΩ$, so for $f\equivλ$ 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 $\{|\nabla u|=σ\}$ of positive measure. When $Λ=\infty$, 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 $f\equiv0$ 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.