Syllabus
General notation of weak solutions
Sobolev spaces: definition and basic overview of its properties, embedding and trace theorems
Weak solutions to linear elliptic equations on bounded domains, various boundary conditions, solution by the use of the Riesz representation theorem and the use of the Lax-Milgram theorem, compactness of the solution operator, eigen-values and eigen-vectors of the solution operator, Fredholm-like theorems and their applications, maximum principle for weak solution, $W^{2,2}$ and higher regularity, symmetric operators and the equivalence with minimizing of a quadratic functional
Bochner spaces: defintion and basic overview of its properties, Gelfand triple, integration by parts formula, embeddding.
Weak solutions to linear parabolic equations, various boundary conditions, construction of a solution via Galerkin method, uniqueness and regularity of solution.
Weak solution to linear hyperbolic equation, various boudary codition, construction of a solution via Galerkin method, uniqueness of solution, finite speed of propagation.
Annotation
This is the basic course about the theory of partial differential equations. The notion of a weak (distributional) solution and the corresponding function spaces will be introduced and we establish the theory for (linear) elliptic equations.