在這篇論文中，主要是要討論一些哈奈克定理(Harnack Theorem)。這些定理是關於均勻的二階橢圓偏微分方程和二階拋物偏微分方程的正解之性質。
我們討論哈奈克不等式分成兩種情形，一種情形是方程中的係數是假設光滑的，另一種情形的係數，則假設為有界和可測的。第一種情形我們使用的方法是梯度估計，而第二種情形我們用莫瑟迭代法(Moser Iteration)來證明。
我們知道哈奈克定理也可以看成是一種直接且量化的極大值原則(Maximum Principle)。哈奈克定理的衍生應用，可以證明橢圓偏微分方程或拋物偏微分方程的解有赫爾德連續(Hölder continuity)的性質。除此之外，由於最小曲面的坐標函數是調和的，因此哈奈克性質也可用來觀察坐標函數的發展。

This thesis is a survey of some Harnack theorems for the positive solutions of uniformly elliptic differential equations of second order and for the positive solutions of parabolic differential equations of second order.
In particular, we shall derive the Harnack’s inequalities for the following two situations separated according to the regularity of the coefficients. The first situation is when the coefficients of the equation are assumed to be smooth, and we obtain the Harnack inequality for the positive solution by using the gradient estimate method. The second situation is when the coefficients are assumed bounded and measurable, and we obtain the Harnack inequality for the positive weak solution by using the Moser iteration method.
Since a Harnack inequality can be viewed as a sharpened and quantitative version of the maximum principle one can apply the Harnack inequality to prove the Hölder continuity for the solutions and to study the growth condition for the coordinate functions of minimal surfaces in the Euclidean space.

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