### 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, lindblad@math.jhu.edu. Office hour: W 3-4 Krieger 406.
TA: Luo, cyluo@math.jhu.edu, 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