On the geometry of the p-Laplacian operator

The p-Laplacian operator Δₚu= div (|∇ u|ᵖ⁻²∇ u) is not uniformly elliptic for any p∈(1,2)∪(2,∞) and degenerates even more when p→ ∞ or p→ 1. In those two cases the Dirichlet and eigenvalue problems associated with the p-Laplacian lead to intriguing geometric questions, because their limits for p→∞ or p→ 1 can be characterized by the geometry of Ω. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general p∈[1,∞]. We report also on results concerning the normalized or game-theoretic p-Laplacian{document}ΔₚNu:=1/p|∇ u|²⁻ᵖΔₚu=1/pΔ₁Nu+p-1/pΔ_∞Nu{document}and its parabolic counterpart uₜ-ΔₚN u=0. These equations are homogeneous of degree 1 and ΔₚN is uniformly elliptic for any p∈ (1,∞). In this respect it is more benign than the p-Laplacian, but it is not of divergence type.