MA6090 Sobolev Spaces and applications to PDE (modified)
Distribution theory: The space of test functions and the convergence, the space of distributions, support and singular support of distributions, the convolution of distributions, fundamental solutions, tempered Distributions. 
Sobolev spaces: Definition and basic properties Sobolev spaces, extension theorems, Sobolev embedding theorems, Rellich-Kondrasov compactness theorems, dual spaces, fractional order Sobolev spaces, trace theorems. 
Weak Solutions: Abstract variational problems, Lax-Milgram theorem, weak solutions of second order elliptic equations, regularity of weak solutions, maximum principle, the eigenvalues of Laplacian. 
1. S. Kesavan, Topics in Functional Analysis and Applications, New Age International Publishers, 2015.
2. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
1. R. A. Adams and J. F. Fournier, Sobolev Spaces, Academic Press, 2003.
2. W. Rudin, Functional Analysis, Tata McGraw-Hill, 2006.
3. R. S. Strichartz, A guide to Distribution Theory and Fourier Transforms, World Scientific, 2008.
4. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004.