The strong maximum principle for Schrödinger operators on fractals

Marius V. Ionescu 1 , Kasso A. Okoudjou 2 ,  and Luke G. Rogers 3
  • 1 Department of Mathematics, United States Naval Academy, Annapolis
  • 2 Department of Mathematics and Norbert Wiener Center, University of Maryland, MD 20742, College Park
  • 3 Department of Mathematics, University of Connecticut, Storrs

Abstract

We prove a strong maximum principle for Schrödinger operators defined on a class of postcritically finite fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.

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