MA545, Spring 2013. Office Hours: Monday and Wednesday from
1:30-3:00 p.m. or by appointment.
The main focus of this class will be the analysis of functions
of Several Variables. We will study differentiablity of functions
of Several Variables, Change of Variable Formula, Sobolev
functions, approximation of Sobolev functions, Sobolev Imbedding
Theorem. Then we will apply these results to the Laplace's
equation to obtain the regularity of weakly harmonic functions. In
order to accomplish these goals, we will need to study Weak and
Strong convergence in L^p, the Riesz Representation Theorem and a
brief introduction to the Theory of Distributions. Other topics
that will be reviewed are Differentiation of measures,
Convolutions, Distribution functions, Covering Theorems.
We will use the following books:
1. Modern Real Analysis, by William Ziemer (with Monica Torres).
2. Measure Theory and Fine Properties of Functions, by Craig
Evans and Ronald Gariepy.
3. Partial Differential Equations, by Craig Evans.