Anisotrope elliptische partielle Differentialgleichungen I consider a prototype of an anisotropic elliptic partial differential equation. At the beginning the necessary definitions and properties of anisotropic Sobolev spaces will be presented followed by some main examples of anisotropic partial differential equations together with statements about existence uniqueness and regularity of its solutions. After this introduction to anisotropic problems some maximum principles will be proved both for weak and viscosity solutions. Finally a Harnack inequality for positive solutions of anisotropic elliptic partial differential equations in IR2 will be proved.