Evans偏微分方程筆記/問題解答/視頻講解

Evans偏微分方程筆記/問題解答/視頻講解

Exercises

2019-2020-2 (2020 上半年) PDE Exercises

5 Sobolev spaces

5.1 Holder spaces

5.2 Sobolev spaces

5.2.1 Weak derivatives

5.2.2 Defintion of Sobolev spaces

5.2.3 Elementary properties

5.3  Approximation

5.3.1 Interior approximation by smooth functions

5.3.2 Approximation by smooth functions

5.3.3 Global approximation by smooth functions

5.4 Extensions

5.5 Traces

5.6 Sobolev inequalities

5.6.1 Gagliardo-Nirenberg-Sobolev inequality

Video Evans 5-6-1-1

5.6.2 Morrey's inequality

Video Evans 5-6-2-1, part 2

Video Evans 5-6-2-3

for a continuous version

5.6.3 General Sobolev inequalities

Video Evans 5-6-3

5.7 Compactness

5.8 Additional Topics

5.8.1 Poincar'e's inequality

Video Evans 5-8-1-1

Video Evans 5-8-1-2

5.8.2 Difference quotients

5.8.2-1 Difference quotients and  

Video Evans 5-8-2-1-1

Video Evans 5-8-2-1-2

5.8.2-2 Lipschitz functions and

5.8.3 Differentiable  

5.8.4 Hardy's inequality

5.8.5 Fourier transform methods

5.9 Other spaces of functions

5.9.1 The space

5.9.2 Spaces involving time

5.10 Problems and Solutions

5.10.1 Holder spaces are Banach spaces

5.10.2 An interpolation inequality for Holder norms

5.10.3 A parameter of Sobolev functions

5.10.4 Sobolev spaces on an open interval

5.10.5 A cut-off function

5.10.6 Partition of unity

5.10.7 A new proof of the trace theorem

5.10.8 No Trace operator for Lebesgue spaces

5.10.9 An interpolation inequality

5.10.10 Two interpolation inequalities

5.10.11 Sobolev functions with weak derivative

5.10.12 An counterexample for difference quotients.

5.10.13 An example showing

5.10.14

5.10.15 A Poincar'e-type inequality

5.10.16 Variant of Hardy's inequality

5.10.17 Chain rule

5.10.18 The positive/negative part of Sobolev functions

5.10.19 Alternative proof of Problem 18 (3)

5.10.20

5.10.21

6 Second-order elliptic equations

6.1 Definitions

6.1.1 Elliptic equations

6.1.2 Weak solutions

6.2 Existence of weak solutions

6.2.1 Lax-Milgram Theorem

6.2.2 Energy estimates

6.2.3-1 Second existence theorem for weak solutions

6.2.3-2 Third existence theorem for weak solutions

6.2.3-3 Boundedness of the inverse

6.6.1 Laplace's equation with potential function

6.6.2 Second-order elliptic operator with no transport

6.6.3 Unique weak solution of the biharmonic function

6.6.4 Weak solutions of Neumann's problem

6.6.5 Unique weak solution of Poisson's equation with Robin

6.6.6 Unique weak solution of Poisson's equation with mixed Dirichlet-Neumann boundary conditions

Appendix

A. Notation

A.1. Notation for matrices

A.2. Geometric noation

A.3. Notation for functions

A.4. Vector-valued functions

A.5. Notation for estimates

A.6. Some comments about notation

B. Inequalities

B.1. Convex functions

B.2. Useful inequalities

B.2.1. Cauchy's inequality

B.2.2 Cauchy's inequality with epsilon

B.2.3. Young's inequality

B.2.4. Young's inequality with epsilon

B.2.5. H"older's inequality

B.2.6. Minkowski's inequality

B.2.7. General H"older inequality

B.2.8. Interpolation inequality for -norms

B.2.9. Cauchy-Schwarz inequality

B.2.10. Gronwall's inequality (differential form)

B.2.11. Gronwall's inequality (integral form)

C. Calculus

C.1. Boundaries

C.2. Gauss-Green theorem

C.3. Polar coordinates, coarea formula

C.4. Moving regions

C.5. Convolution and smoothing

C.6. Inverse function theorem

C.7. Implicit function theorem

C.8. Uniform convergence

D. Functional Analysis

D.1. Banach spaces

D.2. Hilbert spaces

D.3. Bounded linear operaters

D.3.1. Linear operators on Banach spaces

D.3.2. Linear operators on Hilbert spaces

D.4. Weak convergence

D.5. Compact operators, Fredholm theory

D.5.1 Definitions and basic properties

D.5.2. Fredholm alternative

D.5.3. The spectrum of compact operators

D.6. Symmetric operators

D.6.1.Bounds on spectrum of symmetric bounded linear operator

D.6.2.Eigenvectors of a compact,symmetric operator

E. Measure theory

E.1. Lebesgue measure

E.2. Measurable functions and integration

E.3. Convergence theorems for integrals

E.4. Differentiation

E.5. Banach space-valued functions