The aims of the course are twofold. First, we discuss the most important properties of certain vector spaces of weakly differentiable functions whose foundations were laid in the ground-breaking works of S.L. Sobolev (1935,1936). Sobolev spaces naturally arise in a wide range of areas of mathematics and physics and have surprisingly rich mathematical properties. They are especially known for their crucial role in the modern theory of partial differential equations. Second, with this in mind, we employ Sobolev spaces for studying linear elliptic, parabolic, and hyperbolic partial differential equations.
R.A. Adams, J.J.F. Fournier. Sobolev Spaces. Second Edition. Academic Press (2003).
L.C. Evans. Partial Differential Equations. American Mathematical Society (1998).
L.C. Evans, R.F. Gariepy. Measure Theory and Fine Properties of Functions, Revised Edition. Taylor & Francis (2015).
S.L. Sobolev. Some Applications of Functional Analysis in Mathematical Physics. American Mathematical Society (2008).