Math 631. Partial Differential Equations I: Linear Equations
- Spring 12 - Hans Lindblad

Description:
Lectures: MW 1.30-2.45 in Ames 219. Instructor Hans Lindblad, [email protected]. Office hour: W 3-4 Krieger 406.
TA: Luo, [email protected], Office hour: in Krieger 211
Text: Evans Partial Differential Equations
Syllabus:Classical theory of first order systems, the wave, heat and Laplace equations. Existence and uniqueness of solutions of the initial value problem respectively boundary problem. Characteristics, Cauchy-Kowalewski theorem, Fundamental solutions, Fourier methods, the Energy inequality. Existence for elliptic equations with variable coefficdients. The Fourier transform, Sobolev spaces and inequalities, distributions and other methods from functional analysis like Riesz Representations theorem, Compactness, Spectral Theorem, will be developed as needed. We will use material from Evans, Partial Differential Equations", as well as on occasions from the basic PDE books by Strauss, Taylor, John, Rauch, Folland and the text by Reed & Simon, Methods of Mathematical Physics.
Preliminary schedule:
wk  date  Monday  Wednesday  Homework due following Wed.
  1  1/30  InitialValueProblem  Fourier Series   1.1,1.2,2.1,2.2 in lecture notes
  2  2/6  IVP with FS, FourierTransform  FourierTransform, IVP with FT  2.1-2,3.1-2,4.1-4 in LN
  3  2/13  Weak solutions,Distributions  Operations on Distributions  5.1-3, 6.1-6 LN
  4  2/20  Fundamental Solutions  2.2 Harmonic functions  7.1-5 in LN, 2,3,4,6 on p 85-6
  5  2/27 2.2.4 Green's functions.  2.3.1 Heat equation   8,9,11,12a,13,14 p 86-87
  6  3/5  2.4 Wave eq.  2.4  2.5: 18,19, 24 page 88-90.
  7  3/12 3.2 Characteristics  4.6 Analytic Solutions   Evans 4,8 p 162-3, Notes 1,2
  8  3/19  break  break  
  9  3/26  5.1-2 Sobolev Spaces  5.3 Approximation  4,5,6 p 306
10  4/2  5.4 Extensions, 5.5 Traces  5.6 Sobolev Inequalities  
11  4/9  5.6 Sobolev Inequalities  5.7 Compactness 5.8 PoincareIneq  7,8,9,14,15
12  4/16  5.8 Difference quotients  5.9 Other spaces  16,17,20,21
13  4/23  Elliptic PDE 6.1-2 Weak Sol  6.2,App D5 Exist  
14  4/30  6.3 Regularity  6.5, App D 5-6 Eigenvalues  p 365-370:2,4,7,8(read 6.4),13