Consider the p-Laplacian of a suitably nice function $u$:
$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$
Are there useful ways of thinking about the p-Laplace operator, or of thinking about p-harmonic functions (the functions $u$ with $\Delta_p u = 0$ in some domain)?
What are some application of the p-Laplacian that could help in building intuition for it?
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Similar question for the Laplacian: Intuitive interpretation of the Laplacian
Well, in a sense that's only useful for other mathematicians, it's about minimizing the $L^p$ norm of the gradient.
In general, you can think of it as another averaging differential operator: it has the form $\nabla \cdot (D \nabla u)$. In heat transport equations, D is the diffusion coefficient, which in this case is proportional to a power of the gradient.
There is also a detailed explanation of some of the physics background of the equation at Math Overflow here
One practical application of $\int |\nabla u|^p\to \min$ is in image denoising. If $u$ is the light intensity, then noise contributes to $|\nabla u|$. The easiest thing to do computationally is to decrease $\int |\nabla u|^2$, which more or less amounts to Gaussian smoothing. Unfortunately, $\int |\nabla u|^2$ penalizes not only noise, but also sharp edges, which become blurry in the process.
Taking $p=1$ instead of $p=2$, we arrive at TV denoising, which preserves edges. The drawback here is that the total variation functional is not strictly convex (bad for minimum search) and the associated Euler-Lagrange equation $\nabla \cdot (|\nabla u|^{-1}\nabla u)$ is awfully degenerate. One way around this is to minimize $\int \alpha |\nabla u| + |u-u_0|^2$, with $u_0$ being the noisy image. This addition of the quadratic term makes the optimization process more palatable: see here, for example.
On the opposite extreme, for large $p$, the $p$-Laplacian is used to model growing/collapsing sandpiles.
