### MA6090 Sobolev Spaces and applications to PDE (modified)

#### Course Details

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. [12]

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. [16]

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. [12]

#### Course References:

Text Books:

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.

Reference Books:

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.