Fall 2014
Classes are 50 minutes long.
Day 1
1.1 Contraction mapping principle
1.2 Application to Picard’s existence and uniqueness theorem
Day 2
1.2 Application to Picard’s existence and uniqueness theorem, cont.
1.3 Fredholm equation (statement only)
1.4 Contraction-power mapping principle
Day 3
1.5 Application to Volterra’s equation
1.6 Compact operators (definition, and example of nonlinear integral operator)
Over the weekend
Read the first three pages of Chapter 2 (up through Example 2.8), to refresh your memory on the basics of Hilbert space theory.
We will not cover this material in class. You are welcome to ask me about it in office hours or after class.
Day 4
1.6 Compact operators (approximation by finite-rank operators)
1.7 Brouwer and Schauder fixed point theorems
Day 5
1.8 Application to Peano’s existence theorem
2.1 Hilbert space basics (noncompleteness of C^1 with Sobolev inner product)
Day 6
2.1 Hilbert space basics, cont. (Orthogonal Decomposition, proof omitted)
2.1 Hilbert space basics, cont. (Riesz Representation)
Day 7
2.2 Weak solution of Poisson equation
2.3 Orthonormal bases
Day 8
2.3 Orthonormal bases, cont.
2.4 Weak compactness
Day 9
2.4 Weak compactness, cont.
3.1 – Green’s theorem, and integration by parts [assigned reading; not covered in class]
3.2 Mollification and smoothing
Day 10
3.2 Mollification and smoothing, cont.
Day 11
3.3 Weak derivatives and Sobolev spaces
Day 12
3.4 Approximating Sobolev functions by smooth functions
3.5 Sobolev functions vanishing on the boundary
Day 13
3.6 Extending past the boundary
3.7 Boundary traces (statement)
Day 14
3.7 Boundary traces (proof)
3.8 Sobolev inequalities
Day 15
3.8 Sobolev inequalities, cont.
Day 16
3.8 Sobolev inequalities, cont.
Day 17
3.9 Compact imbedding of Sobolev spaces (Rellich-Kondrachev)
Day 18
3.9 Compact imbedding of Sobolev spaces, cont.
3.10 Application: Poisson’s Equation via Calculus of Variations
Day 19
3.10 Application: Poisson’s Equation via Calculus of Variations, cont.
4.2 Spectral theorem for compact, selfadjoint operators (statement only)
Day 20
4.1 Abstract spectral theory for sesquilinear forms
Day 21
4.1 Abstract spectral theory for sesquilinear forms, cont.
Day 22
4.3 Application to elliptic operators
Day 23
5.1 Generalized Poisson equation – wellposedness
Day 24
5.1 Generalized Poisson equation, cont.
5.2 Regularity of solutions
Day 25
5.2 Regularity of solutions, cont.
Day 26
5.2 Regularity of solutions, cont.
5.3 Weak Maximum Principles
Day 27
5.3 Weak Maximum Principles, cont.
5.4 Strong Maximum Principles
Day 28
5.4 Strong Maximum Principles, cont.
Then discuss non-symmetric elliptic PDEs via the Lax-Milgram Theorem
Day 29
Non-symmetric elliptic PDEs and Lax-Milgram, cont.
Day 30
6.1, 6.2 Parabolic equations and the Galerkin approximation
Day 31
6.2 cont. and 6.3 Parabolic equations and the Galerkin approximation
Day 32
6.4 Energy estimates and weak solutions
Day 33
6.5 Maximum principles
Day 34
7.1, 7.2 Hyperbolic equations
Day 35
7.3 Finite speed of propagation
Day 36
8.1 Generators, Resolvents
Day 37
8.1 Generators, Resolvents, cont.
Day 38
8.2 Statement of Hille-Yosida, and sketch of proof
Day 39
8.3 Dissipative operators
Day 40
8.3 Dissipative operators, cont.
Day 41
8.4 Application: solving parabolic and hyperbolic PDEs by semigroups
Day 42
8.5 Application: nonhomogeneous and nonlinear evolution equations
Day 43
To be determined
Fall 2013
Classes are 80 minutes long, with a 5 minute break in the middle.
Day 1
1.1 Contraction mapping principle
1.2 Application to Picard’s existence and uniqueness theorem
Day 2
1.3 Fredholm equation (statement only)
1.4 Contraction-power mapping principle
1.5 Application to Volterra’s equation
1.6 Compact operators (definition, and example of nonlinear integral operator)
Day 3
1.6 Compact operators (approximation by finite-rank operators)
1.7 Brouwer and Schauder fixed point theorems
1.8 Application to Peano’s existence theorem
Day 4
2.1 Hilbert space basics (inner products, noncompleteness of C^1 with Sobolev inner product)
Day 5
2.1 Hilbert space basics (Orthogonal Decomposition and Riesz Representation)
Day 6
2.2 Weak solution of Poisson equation
2.3 Orthonormal bases
Day 7
2.4 Weak compactness
3.2 Mollification and smoothing
Day 8
3.3 Weak derivatives and Sobolev spaces
Day 9
3.4 Approximating Sobolev functions by smooth functions
3.5 Sobolev functions vanishing on the boundary
3.6 Extending past the boundary
Day 10
3.7 Boundary traces
3.8 Sobolev inequalities
Day 11
3.8 Sobolev inequalities, continued
Day 12
3.9 Compact imbedding of Sobolev spaces (Rellich-Kondrachev)
Day 13
3.10 Application: Poisson’s Equation via Calculus of Variations
Day 14
4.1 Abstract spectral theory for sesquilinear forms
Day 15
4.2 Spectral theorem for compact, selfadjoint operators
Day 16
4.2 Spectral theorem, continued
4.3 Application to elliptic operators
Day 17
5.1 Generalized Poisson equation – wellposedness
Day 18
5.2 Regularity of solutions
Day 19
5.2 Regularity, cont.
5.3 Weak Maximum Principles
Day 20
5.4 Strong Maximum Principles
Day 21
6.1, 6.2, 6.3 Parabolic equations and the Galerkin approximation
Day 22
6.4 Energy estimates and weak solutions
6.5 Parabolic maximum principles
Day 23
7.1, 7.2 Hyperbolic equations
Day 24
7.3 Finite speed of propagation
8.1 Semigroups
Day 25
8.1 Generators, Resolvents
8.2 Statement of Hille-Yosida
Day 26
8.2 Proof of Hille-Yosida
8.3 – Dissipative operators
Day 27
8.3 Dissipative operators, cont.
8.4 Application: solving parabolic and hyperbolic PDEs by semigroups
Day 28
8.4 Application: solving parabolic and hyperbolic PDEs by semigroups, cont.
Day 29
To be determined