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

We characterize solutions of the regularized $p$-Laplace equation $$\mathrm{div}\text{\hspace{0.17em}⁣}((1+\mathrm{\mid }Dv{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-1}Dv)=0,1<p<\mathrm{\infty },$$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ by a pointwise asymptotic mean value identity. For $v\in C^2(\mathrm{\Omega })$, solving the equation is equivalent to $$v(x)=\frac{\tilde{\alpha }}{2}({\mathcal{S}}_{\epsilon }^{+}[v](x)+{\mathcal{S}}_{\epsilon }^{-}[v](x))+\tilde{\beta }{\int }_{B_{\epsilon }(0)}v(x+h){\rho }_{\epsilon }(h)dh+o({\epsilon }^2),$$ where $$\tilde{\alpha }=\frac{p-2}{p+n+1},\tilde{\beta }=\frac{n+3}{p+n+1}\mathrm{.}$$ 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 $$w(x,s)=v(x)+s\mathrm{.}$$ 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 $1<p<\mathrm{\infty }$. 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}.