Anisotropic p-Laplacian Evolution of Fast Diffusion type

Abstract

We study an anisotropic, possibly non-homogeneous version of the evolution p-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive L 1 to L ∞ estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

The concentration-compactness principle for the nonlocal anisotropic $p$-Laplacian of mixed order

“…, p n > 1. This operator can be seen as a nonlocal analog of the anisotropic p-Laplacian Local operators with such an orthotropic structure are well known in the literature and there are several results related to this type of operators, see [36,41,8,20,21,26,10,16,22,24,11] and the references therein. We can read off (1.1), that the operator under consideration has on the one hand different exponents of integrability and on the other hand different orders of differentiability.…”