2. On the p-Laplace equation and the p-parabolic equation

The following six papers deal with the p-Laplace (or p-harmonic) equation in the plane:

• Construction of singular solutions to the p-harmonic equation and its limit equation for $p=\infty $.
  Manuscripta Mathematica 56(1986), 135-158.
• On certain p-harmonic functions in the plane.
  Manuscripta Mathematica 61(1988), 79-101.
• On p-harmonic Functions in the Plane and Their Stream Functions,
  by G. A. and P. Lindqvist,
  J. Diff. Eq. 74:1 (1988), 157-178.
• Representation of a p-harmonic function near a critical point in the plane,
  Manuscripta Mathematica 66(1989), 73-95.
• Aspects of p-harmonic Functions in the Plane,
  Lecture notes from the Finnish Summer School in Potential Theory 1990, pp.9-34. Published by the University of Joensuu 1992. Editor I. Laine.
• On a nonlinear cousin of the Laplace equation. Published by Uppsala University 1995, in "Festschrift in Honour of Lennart Carleson and Yngve Domar", pp.117-123. Editor A.Vretblad.

The fourth of these papers contains a complete determination of the optimal regularity (Hölder smoothness of the gradient) for p-harmonic functions in the plane. The "p-Cauchy-Riemann equations" were introduced in the second of the above papers.

The p-Laplace equation, including the p-Poisson equation, has some applications in continuum mechanics as seen from the following three papers:

• On Hele-Shaw flow of power-law fluids, by G. A. and U. Janfalk, Euro. Jnl. of Appl. Math. 3(1992), 343-366.
• On p-harmonic functions, convex duality and an asymptotic formula for injection mould filling, Euro. Jnl. of Appl. Math. 7(1996), 417-437.
• Stationary pipe flow of power-law fluids, Proceedings of the Fifth European Conference on Mathematics in Industry, 135-138. Teubner 1991. Editor M. Heiliö.

By the p-parabolic equation we mean the time-dependent version of the p-Laplace equation. We write it as $$u_t = div(|\nabla u|^{p-2}\nabla u)+F(x,t).$$

The departmental report

• Remarks and conjectures for the p-parabolic equation, Linköping 1995,

contains some discussion of the physical meaning of this equation, some geometric analysis and conjectures concerning the behaviour of solutions when $p\to \infty$. In particular, a possible relation to a system describing the development of a collection of sand cones is conjectured. This relation was rigorously established, among many other things, in the paper

• Fast/Slow Diffusion and Growing Sandpiles, by G. A. and C. Evans, J. Diff. Eq.131:2(1996), 304-335.