A DIFFERENTIAL EQUATION FRAMEWORK ON ANISOTROPIC FRACTAL POROUS MEDIA I: THE SCALAR CASE

In this paper, we develop a PDE framework on anisotropic fractal structures, which are fractal materials that could have different dimensions in each spatial direction. In particular, we develop the notion of distributional fractal gradient and the appropriate weighted Sobolev spaces for an equivalent continuum problem. We prove a number of related results, such as a new Poincaré Lemma and Poincaré inequality. One of the main results is a new scalar trace theorem, which allows for the solvability of the fractal Laplace and Poisson equations with non-homogeneous boundary data. Along the way, we track the dependence in the various estimates on the fractal dimension in each direction and the length of the original fractal solid.