the p-Laplacian
The ordinary Laplacian is the operator behind the most efficient, lowest-energy configurations — a soap film, an electric potential, steady heat. It is the gradient of the simplest energy, the integral of (1/2)|grad u|^2. But what if the energy you want to minimize measures the gradient with a different power? Replace the exponent 2 by a general p, and the operator you get is the p-Laplacian — a one-parameter nonlinear generalization that interpolates between very different behaviours and shows up wherever the 'cost of a gradient' is not quadratic.
It is defined as Delta_p u = div( |grad u|^(p-2) grad u ), for a parameter p > 1. When p = 2 the factor |grad u|^0 = 1 and it collapses to the ordinary Laplacian div(grad u) = Laplacian u. For other p it is genuinely nonlinear: the coefficient |grad u|^(p-2) depends on how steep u is. It is exactly the Euler-Lagrange operator of the energy integral of (1/p)|grad u|^p, so 'p-harmonic' functions (those with Delta_p u = 0) are the minimizers of that p-energy — the natural notion of 'as flat as possible' when flatness is measured in the L^p sense. The parameter changes the personality: for large p the operator strongly penalizes steep gradients (the limit p to infinity links to the infinity-Laplacian and distance functions); for p near 1 it is close to total-variation flow, used in image denoising; the time-dependent version u_t = Delta_p u is a degenerate or singular nonlinear diffusion akin to the porous-medium equation.
Why study it? The p-Laplacian is the standard testbed for nonlinear elliptic and parabolic theory: it is the simplest operator that is nonlinear yet still variational and degenerate-elliptic, so it is where regularity techniques for nonlinear PDEs are honed (p-harmonic functions are not smooth in general — only C^(1,alpha) — which already signals that nonlinearity costs you regularity). It appears in non-Newtonian fluids, glaciology, plasticity, and image processing. The honest point: the single tweak from exponent 2 to exponent p turns a clean linear theory into a genuinely nonlinear one, with degeneracy where grad u = 0 and only limited smoothness — a controlled laboratory for everything that goes wrong (and can be fixed) in nonlinear ellipticity.
Take p = 4: then Delta_4 u = div(|grad u|^2 grad u), the Euler-Lagrange operator of the energy integral of (1/4)|grad u|^4. Minimizing this energy with fixed boundary values gives a '4-harmonic' function — the flattest configuration when steepness is penalized to the fourth power, which suppresses large gradients far more aggressively than the ordinary (p = 2) harmonic function does.
Replace exponent 2 by p: a one-parameter nonlinear Laplacian.
Unlike harmonic functions, p-harmonic functions (p not equal to 2) are generally NOT smooth — the best you get is C^(1,alpha) regularity, and the operator degenerates where grad u = 0. The single exponent change buys genuine nonlinearity at the cost of guaranteed smoothness.