Harnack inequality for elliptic equations in non-divergence form

This thesis deals with Harnack Inequality for Non-divergence Elliptic Type Partial Differential Equations (PDEs) which takes an important part in the theory of PDEs. The thesis consists of four chapters and the reference list.In the first chapter, the progress and the usage area of Harnack Inequality have been discussed. In the second chapter, the fundamental definition used in other chapters and related information have been given and the uniqueness solution of Dirichlet Problem in a bounded domain has been viewed. In the third chapter, by examining the Harnack Inequality for harmonic functions and applying known methods, Harnack Inequality has been proved. In the last chapter, weak and strong maximum principles which take an important place to show the eniqueness solution of Dirichlet problem, related to subject, have been examined. Disputing the Normal Derivative Lemma for the proof of strong maximum problem and Aleksandrov type maximum principle, ?s-capacity?, ?Growth lemma? which are very useful to understand Harnack Inequality have been discussed. Furthermore, it has been tried to explain the importance of the growth lemma for this area and the proof of the general form of the growth lemma for non-divergence PDEs which was suggested by Krylov and Safanov has been prospected. The last part of the thesis consists of some main theorems of current articles.