An Asymptotic Mean Value Characterization for the Regularized $p$-Laplacian

Abstract

We characterize solutions of the regularized $p$-Laplace equation \[ \operatorname{div}\!\left((1+|Dv|^2)^{p/2-1}Dv\right)=0, \qquad 1<p<\infty, \] in a bounded domain $\Omega \subset {\mathbb{R}}^n$ by a pointwise asymptotic mean value identity. For $v\in C^2(\Omega )$, solving the equation is equivalent to \[ v(x) = \frac{\widetildeα}{2} \left( \mathcal{S}_{\varepsilon}^{+}[v](x) + \mathcal{S}_{\varepsilon}^{-}[v](x) \right) + \widetildeβ \int_{B_\varepsilon(0)} v(x+h)ρ_\varepsilon(h)\,dh + o(\varepsilon^2), \] where \[ \widetildeα = \frac{p-2}{p+n+1}, \qquad \widetildeβ = \frac{n+3}{p+n+1}. \] The kernel ${\rho }_{\epsilon }$ is the semicircular marginal of normalized Lebesgue measure on the $(n+1)$-dimensional ball, and ${\mathcal{S}}_{\epsilon }^{+}$ and ${\mathcal{S}}_{\epsilon }^{-}$ 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 $1<p<\infty$. 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 \cite{JLM01}. The convergence of the associated projected dynamic programming scheme is established in the companion paper \cite{Moosavi26}.