__上學期__

兩份重要資料：

__課程大綱__、

__預計課程進度__（僅供參考）

__Week 1__: 1. Overview of partial differential equations; 2. Harmonic functions; 3. Mean Value Theorem; 4. Interior estimates, Gradient estimates; 5. Harnack’s inequality.

__Week 2__: 1. Green’s formulas; 2. The representation formula of a harmonic function; 3. Green’s function; 4. The Poisson kernel; 5. The Poisson representation.

__Week 3__: No class.

Homework #1, 1.1-1.9 (due on 10/6).

__Week 4__: 1. Characterization of a harmonic function by the mean value property; 2. Weyl’s theorem.

__Week 5__: 1. Subharmonic functions; 2. Strong and weak maximum principles; 3. Removable singularity; 4. Gradient estimates of a harmonic function; 5. Liouville’s theorem; 6. 2nd order elliptic equations; 7. Weak maximum principle.

__Week 6:__1. Weak maximum principle with sign condition; 2. Strong maximum principle of E. Hopf; 3. Hopf’s boundary lemma; 4. ABP estimates.

Homework #2, 2.1-2.5 (due on 10/27).

* Midterm is scheduled on 11/17 (Tue).

__Week 7:__1. Proof of ABP estimate; 2. Application of ABP method; 3. Comparison principles for fully nonlinear elliptic equations.

Homework #3, 2.6-2.11 (due on 11/3).

__Week 8:__1. Maximum principles for fully nonlinear elliptic equations; 2. Maximum principle for narrow domains; 3. Refined ABP estimate.

__Week 9:__1. Moving plane method; 2. Perron’s method.

__Week 10:__1. Proof of Perron’s method; 2. Barrier; 3. Regularity of the boundary.

__Week 11:__1. Sufficient condition for the regularity; 2. Exterior sphere property; 3. The heat equation; 4. Parabolic boundary; 5. The maximum principle; 6. The maximum principle in the whole space; 7. Uniqueness of the initial value problem of the heat equation.

Homework #4, 3.5, 4.1 (due on 12/1).

__Week 12:__1. Exponential growth condition; 2. Parabolic Harnack’s inequality; 3. Strong maximum principle; 4. Fundamental solution of the heat operator; 5. The solution of the initial value problem of the heat equation.

__Week 13:__1. The initial boundary value problem for the heat equation; 2. Potential theory; 3. Jump relation.

__Week 14:__No classes.

__Week 15:__1. Solving the initial boundary value problem for the heat equation; 2. The fixed-point integral equation; 3. The heat kernel; 4. The initial boundary value problem with inhomogeneous term; 5. The Duhamel principle.

Homework #5, 4.2, 4.3, 4.4 (due 12/29).

__Week 16:__1. The convergence of the heat flow; 2. Dirichlet boundary value problem.

__Week 17:__1. Reaction-diffusion equations; 2. The existence and uniqueness theorem; 3. The fixed point method; 4. Gronwall type inequality.

* Final exam is scheduled on 1/5 (Tue).

__下學期__

課程訊息：

__課程大綱__、

__預計課程進度__

__Week 1__: 1. Traveling wave solutions of the reaction-diffusion equation; 2. Reaction-diffusion systems; 3. Turing mechanism (instability).

__Week 2__: 1. Wave equations; 2. Representation formula of solutions of the one-dimensional wave equation; 3. Finite speed of propagation; 4. Domain of influence, Domain of dependence; 5. The method of separation of variables; 6. Conservation of energy.

作業題號請以學校圖書館電子版（第三版）為準

Homework #1 (due on 3/15), 6.3, 6.4, 7.1, 7.2, 7.3.

__Week 3__: 1. Method of spherical means; 2. Representation formula of the solution of the wave equation in 3 dimensions; 3. Huygen’s principle; 4. Method of descent; 5. Representation formula of the solution of the wave equation in 2 dimensions; 6. First order partial differential equations; 7. Characteristic curves.

__Week 4__: 1. Cauchy problems for linear first order equations; 2. Cauchy problems for quasi-linear first order equations; 3. Singularities; 4. Nonlinear first order partial differential equations; 5. Characteristic equations; 6. Strip conditions; 7. Cauchy problems for nonlinear first order partial differential equations.

Homework #2 (due on 3/24), see problems in F. John’s book. Chap 1, Sec 6: 1(a),(b),(c),(d), 2, Section 9: 1,2,3.

__Week 5__: 1. Solving Cauchy problems for nonlinear first order partial differential equations; 2. Characteristic equations; 3. Strip condition; 4. Eikonal equation; 5. Conservation laws; 6. Weak solutions; 7. Rankine-Hugoniot condition; 8. Shocks; 9. Entropy conditions.

Homework #3 (due on 3/31), see problems in F. John’s book. Chap 1, Sec 6: 2,3,4,5,6.

__Week 6__: 1. The heat kernel; 2. Continuous semigroups; 3. The infinitesimal generator of the continuous semigroup A; 4. The domain of A, D(A); 5. Contraction semigroups; 6 D(A) is dense in the Banach space B; 7. The derivative of a contraction semigroup; 8. The resolvent.

Homework #4 (due on 4/7) 8.1, 8.2, 8.3. (Jost’s book)

4/19 Midterm!

__Week 7__: 1. The resolvent identity; 2. Examples of contraction semigroups (translation group, heat group); 3. The Hille-Yosida theorem.

__Week 8__: 1. Proof of the Hille-Yosida theorem; 2. Translation groups; 3. Markov processes; 4. Chapman-Kolmogorov equation.

__Week 9__: 4/19, 10:20-12:20, Midterm. 1. Spatial homogeneity; 2. Brownian motions; 3. Lindeberg’s condition; 4. Contraction semigroup.

__Week 10__: No classes.

__Week 11__: 1. The infinitesimal generator of a Brownian motion; 2. Sobolev spaces; 3. Weak derivatives; 4. W^k,p spaces; 5. Approximation by smooth functions.

__Week 12__: 1. Extensions; 2. The chain rule; 3. Trace map; 4. Characterization of W^k,p_0; 5. Sobolev spaces defined by the Fourier transform; 6. Tempered distributions.

Homework #5 (due on 5/19) Please refer to Evans’ book, Problem section 5.10: 2, 4, 5, 6, 7, 8, 9, 10.

__Week 13__: 1. Coordinates transforms of Sobolev functions; 2. Sobolev spaces on manifolds; 3. Trace map; 4. Sobolev embedding theorems; 5. Gagliardo-Nirenberg-Sobolev inequality (n>p).

__Week 14__: 1. Gagliardo-Nirenberg-Sobolev inequality (n>p); 2. The critical case p=n; 3. Morrey’s inequality (p>n); 4. General Sobolev inequalities.

Homework #6 (due on 6/2) Evans’ book, Problem section 5.10: 13, 14, 15, 16.

__Week 15__: 1. Compact embedding theorems; 2. Negative order of Sobolev spaces; 3. Second order elliptic equations; 4. Weak solutions; 5. Existence of weak solutions; 6. Lax-Milgram theorem.

__Week 16__: 1. Proof of the Lax-Mailgram theorem; 2. Existence and uniqueness of the weak solution to the Dirichlet boundary value problem; 3. Fredholm alternative; 4. Eigenvalue problems.

Final on 6/14, 10:20-12:30.