Evans偏微分方程笔记/问题解答/视频讲解

2020/05/05

张祖锦小锦教学 6天前

Evans偏微分方程笔记/问题解答/视频讲解

Exercises
5 Sobolev spaces
6 Second-order elliptic equations
Appendix

Exercises

2019-2020-2 (2020 上半年) PDE Exercises

5 Sobolev spaces

5.1 Holder spaces Video Evans 5-1

5.2 Sobolev spaces

5.2.1 Weak derivatives Definition and uniqueness of weak derivatives Video Evans 5-2-1-1Examples of existence and nonexistence of weak derivatives Video Evans 5-2-1-2

5.2.2 Defintion of Sobolev spaces Sobolev spaces Video Evans 5-2-2-1Examples: Sobolev functions possess certain smoothness properties, but may stilly badly behaved Video Evans 5-2-2-2

5.2.3 Elementary properties Properties of weak derivatives Video Evans 5-2-3-1 Sobolev spaces are Banach spaces Video Evans 5-2-3-2

5.3 Approximation

5.3.1 Interior approximation by smooth functions Video Evans 5-3-1

5.3.2 Approximation by smooth functions Video Evans 5-3-2

5.3.3 Global approximation by smooth functions Video Evans 5-3-3

5.4 Extensions Video Evans 5.4

5.5 Traces Trace Theorem Video Evans 5.5-1 Trace zero functions in : Video Evans 5-5-2-2

5.6 Sobolev inequalities

5.6.1 Gagliardo-Nirenberg-Sobolev inequality GNS inequality:

Video Evans 5-6-1-1 defined on is an example showing for Video Evans 5-6-1-2

5.6.2 Morrey's inequality Proof part 1

Video Evans 5-6-2-1, part 2 Video Evans 5-6-2-2 For (or more generally, ), , , we have

Video Evans 5-6-2-3 Estimates for ,

for a continuous version of Video Evans 5-6-2-4

5.6.3 General Sobolev inequalities

Video Evans 5-6-3

5.7 Compactness part 1 extension, so that with no boundary trouble; and Video Evans 5-7-1 part 2 fixed 0" data-formula-type="inline-equation" style="margin: 0px; padding: 0px; max-width: 100%; box-sizing: border-box !important; word-wrap: break-word !important;">, is precompact in ; and then 0, \exists\ m_j,\st \vls{j,k} \sen{u_{m_j}-u_{m_k}}_{L^q(V)}\leq \del" data-formula-type="inline-equation" style="margin: 0px; padding: 0px; max-width: 100%; box-sizing: border-box !important; word-wrap: break-word !important;">; later, a standard diagonal argument leads to the desired extraction of a convergent subseuqnece in Video Evans 5-7-2

5.8 Additional Topics

5.8.1 Poincar'e's inequality Poincare inequality Assume is bounded, connected and open, with boundary. Then

Video Evans 5-8-1-1 Poincare inequality for a ball Assume . Then

Video Evans 5-8-1-2 For , Video Evans 5-8-1-3

5.8.2 Difference quotients

5.8.2-1 Difference quotients and For ,

Video Evans 5-8-2-1-1 For ,

Video Evans 5-8-2-1-2

5.8.2-2 Lipschitz functions and Assume is bounded and open, with . Then . Video Evans 5-8-2-2

5.8.3 Differentiable Let . Then function is differentiable Video Evans 5-8-3

5.8.4 Hardy's inequality Video Evans 5-8-4

5.8.5 Fourier transform methods Video Evans 5-8-5

5.9 Other spaces of functions

5.9.1 The space Video Evans 5-9-1

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. case

5.10.13 An example showing

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 \f{n}{2}\ra H^{s(\bbR^n)\subset} L^{\infty(\bbR^n)"} data-formula-type="inline-equation" style="margin: 0px; padding: 0px; max-width: 100%; box-sizing: border-box !important; word-wrap: break-word !important;">

5.10.21 \f{n}{2}\ra H^{s(\bbR^n)"} data-formula-type="inline-equation" style="margin: 0px; padding: 0px; max-width: 100%; box-sizing: border-box !important; word-wrap: break-word !important;">is a Banach algebra

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 Video Evans A1

A.2. Geometric noation Video Evans A2

A.3. Notation for functions Video Evans A3

A.4. Vector-valued functions Video Evans A4

A.5. Notation for estimates Video Evans A5

A.6. Some comments about notation Video Evans A6

B. Inequalities

B.1. Convex functions Video Evans B1

B.2. Useful inequalities

B.2.1. Cauchy's inequality

B.2.2 Cauchy's inequality with epsilon Video Evans B2-1,2

B.2.3. Young's inequality

B.2.4. Young's inequality with epsilon Video Evans B2-3,4

B.2.5. H"older's inequality

B.2.6. Minkowski's inequality Video Evans B2-5,6

B.2.7. General H"older inequality

B.2.8. Interpolation inequality for -norms Video Evans B2-7,8

B.2.9. Cauchy-Schwarz inequality Video Evans B2-9

B.2.10. Gronwall's inequality (differential form) Video Evans B2-10

B.2.11. Gronwall's inequality (integral form) Video Evans B2-11

C. Calculus

C.1. Boundaries Video Evans C1

C.2. Gauss-Green theorem Video Evans C2

C.3. Polar coordinates, coarea formula Video Evans C3

C.4. Moving regions Video Evans C4

C.5. Convolution and smoothing Video Evans C5

C.6. Inverse function theorem Video Evans C6

C.7. Implicit function theorem Video Evans C7

C.8. Uniform convergence Video Evans C8

D. Functional Analysis

D.1. Banach spaces Video Evans D1

D.2. Hilbert spaces Video Evans D2

D.3. Bounded linear operaters

D.3.1. Linear operators on Banach spaces Video Evans D3-1

D.3.2. Linear operators on Hilbert spaces Video Evans D3-2

D.4. Weak convergence Video Evans D4

D.5. Compact operators, Fredholm theory

D.5.1 Definitions and basic properties Video Evans D5-1

D.5.2. Fredholm alternative Video Evans D5-2-1 is closed Video Evans D5-2-2 Video Evans D5-2-3 Video Evans D5-2-4 Video Evans D5-2-5

D.5.3. The spectrum of compact operators Video Evans D5-3-1 Video Evans D5-3-2 is finite or a sequence tending to Video Evans D5-3-3

D.6. Symmetric operators

D.6.1.Bounds on spectrum of symmetric bounded linear operator Video Evans D6-1-1 Video Evans D6-1-2

D.6.2.Eigenvectors of a compact,symmetric operator and their dimensions Video Evans D6-2-1 are orthogonal Video Evans D6-2-2 (the linear span of is dense in Video Evans D6-2-3 construction of a countable orthonormal base Video Evans D6-2-4

E. Measure theory

E.1. Lebesgue measure Video Evans E1

E.2. Measurable functions and integration Video Evans E2

E.3. Convergence theorems for integrals Video Evans E3

E.4. Differentiation Video Evans E4

E.5. Banach space-valued functions Video Evans E5

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