2026/07/03 by Behrooz Moosavi Ramezanzadeh
#math.AP
We characterize solutions of the regularized p-Laplace equation div ((1+|Dv|2)p/2-1Dv)=0, 1<p<∞, in a bounded domain Ω⊂ℝn by a pointwise asymptotic mean value identity. For v∈ C2(Ω), solving the equation is equivalent to v(x) = (\widetildeα)/(2) ( Sε+[v](x) + Sε-[v](x) ) + \widetildeβ ∫Bε(0) v(x+h)ρε(h) dh + o(ε2), where \widetildeα = (p-2)/(p+n+1), \widetildeβ = (n+3)/(p+n+1). The kernel ρε is the semicircular marginal of normalized Lebesgue measure on the (n+1)-dimensional ball, and Sε+ and Sε- are the tilted strategic functionals arising from the affine lift w(x,s)=v(x)+s. The lifted gradient (Dv,1) never vanishes, so the extremal second-order expansion is valid at every gradient regime. The characterization holds for the full range 1<p<∞. By standard interior regularity for nondegenerate regularized p-growth equations, weak solutions are smooth in the interior; the weak and viscosity viewpoints for related quasilinear p-Laplace equations are connected in \citeJLM01. The convergence of the associated projected dynamic programming scheme is established in the companion paper \citeMoosavi26.