Some Geometric Motivations

Let (M, 〈 , 〉) be a complete Riemannian manifold of dimension m ≥ 2, and let Δφ be a quasilinear operator depending on a real function φ, to be detailed in the next chapter. For instance, the family Δφ includes the Laplace–Beltrami operator Δ, defined with the sign agreement that $$\displaystyle \Delta u = \mathrm {div}(\nabla u) = \sum _{j=1}^m \frac {\partial ^2u}{\partial x_j^2} \qquad \text{on } \, \mathbb R^m. $$