In this dissertation, we study aspects of integro-differential operators, and how they relate to different types of equations. In each case, we use information and results about the operators in a lower dimension to analyse an equation in a higher dimension, and vice-versa. We begin in chapter 1 with an introduction to the operators and equations we will be considering.In Chapters 2 and 3, we discuss certain integro-differential operators of functions in a relatively smooth space. However, to understand more about the structure of these operators, particularly about the measure associated with them, we study certain equations in a higher dimension such as degenerate elliptic equations in the upper half space. We analyse the solution of such an equation and its gradient, followed by estimates on its Green's function and Poisson kernel. These estimates then help reveal some properties of the measure associated with the integro-differential operator in the lower dimension. The structure of the degenerate elliptic equations is similar to that of uniformly elliptic equations, but with an additional complexity of a term which involves distance to the boundary. This degeneracy complicates the analysis; as such, the classical techniques of finding pointwise estimates as mentioned above do not work so well anymore. So we provide some revised results for the same. Thus understanding an equation in a higher dimension gives us information about an integro-differential operator in a lower dimension.In Chapters 4 and 5, we prove some results about the solutions of free boundary problems in Rn+1 x [0, T], where the free boundary for a fixed time t can be seen as the graph of a function over a sphere. This time, we connect the solution of the free boundary problem to the solution of a parabolic equation on the sphere - that is, in a lower dimension. This parabolic equation involves an integro-differential operator, which has a min-max representation that is consistent with all the results about viscosity solutions of parabolic equations in Rn. We modify these results for parabolic equations on the sphere, which then gives us existence and uniqueness results about the free boundary problem in a higher dimension.
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