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

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 adapted to the projected tug-of-war construction of \cite{Moosavi26}. For $v\in C^2(\Omega)$, solving the equation is equivalent to $$v(x) = \frac{\widetilde{\alpha}}{2} \left( \mathcal{S}_{\varepsilon}^{+}[v](x) + \mathcal{S}_{\varepsilon}^{-}[v](x) \right) + \widetilde{\beta} \int_{B_\varepsilon(0)} v(x+h)\rho_\varepsilon(h)\,dh + o(\varepsilon^2),$$ where $$\widetilde{\alpha} = \frac{p-2}{p+n+1}, \qquad \widetilde{\beta} = \frac{n+3}{p+n+1}.$$ The kernel $\rho_\varepsilon$ is the semicircular marginal of normalized Lebesgue measure on the $(n+1)$-dimensional ball, and $\mathcal{S}_{\varepsilon}^{+}$ and $\mathcal{S}_{\varepsilon}^{-}$ 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 strategic second-order expansion is valid in every gradient regime, and the characterization holds throughout the full range $1<p<\infty$. Because the normalized equation is uniformly elliptic and the flux has regularized $p$-growth, continuous weak solutions are smooth in the interior, and the identity holds pointwise for the solution itself. For $p\ge 2$, we also prove that the exact projected dynamic programming solutions converge, as $\varepsilon\to0$, to the unique viscosity solution of the regularized Dirichlet problem. The proof uses an amplitude-dependent strict exterior barrier, locally uniform consistency, and the method of half-relaxed limits.