Math 231A. Partial Differential Equations: Fundamental Solutions for the Wave, Heat and Laplace Equations. First order equations. - Fall 07 - Hans Lindblad

Partial differential equations describe many phenomena in Physics such as fluid and heat flow and wave propagation. They are also used in Engineering and Economics. PDE also occur in other branches of mathematics such as geometry, complex analysis and probability. Questions about PDE motivated many of the developments in real and functional analysis. These days one can sometimes numerically solve a PDE approximately, but even so knowledge of the theory is essential. The first question to ask about a PDE is if it has a unique solution and if this solution depends continuously on initial conditions. Then we are also interested in how the solution behaves. The techniques range from abstract functional analytic existence theorems to concrete formulas for and estimates of the solution.

We will follow Evans 'Partial Differential Equations' complemented with lecture notes (and some material from other PDE texts by Folland, John, Rauch, Taylor, Hörmander). The tentative schedule is: In the first quarter I will go over chapters 1-4 in Evans which deal with fundamental solutions for the classical linear constant coefficient Wave, Heat and Laplace equations and general first order equations. In the second quarter Kate Okikiolu will go over chapters 5-6 which deal with Sobolev spaces and variable coefficient (and nonlinear) Elliptic equations (of which the Laplace equation is an example). In the third quarter Jacob Sterbenz will go over chapter 7 about variable coefficient (and nonlinear) parabolic and hyperbolic equations (of which the heat and wave equations respectively are examples)

There will be no exams but some homework problems to be handed in at the end of the quarter. Problem sessions on Fridays 3-4 in AP&M 7218. Lecture notes and assignments will be posted in the schedule.

wk  date  Monday  Wednesday  Friday
  0  9/24  No Class  No Class  PDE, Inital value problem
 1D Transport eq., Wave eq.
  1  10/1  Fourier series solutions  The Fourier transform  Initial value problem with the  Fourier transform
  2  10/8  Distributions  Operations on distributions  Laplace equation:
 2.2.1 Fundamental solution
  3  10/15  2.2.2-3 Meanvalue property
 Max principle, Interior regularity 		 2.2.4 Green's functions,
 2.2.5 Energy methods		  Heat 2.3.1 Fundamental solution  2.3.4 Energy
  4  10/22  No Class  No Class  No Class
  5  10/29  2.4 Wave equation  2.4 Wave equation  3.2 Characteristic curves
  6  11/5  3.2 First order equations  4.6 Noncharacteristic surface  Power series solutions  Hyperbolicity-wellposedness of
 the initial value problem
  7  11/12  Holiday  4.5.3 Geometric Optics Approx.  4.5.3 Stationary Phase
  8  11/19  Hyperbolic versus Elliptic  Propagation of disturbances?  No Class  Holiday
  9  11/26  4.2-3 Elliptic Geometric Optics
 4.5 Elliptic Regularity 		 2.1 Energy estimates for
 symmetric hyperbolic systems		  1.3,5.2 Geometric optics for the wave equation.
 10  12/3  5.3 Geometric optics for  symmetric hyperbolic systems  5.4 Transport equation
 Appendix Eikonal equation		  5.5 Propagation of singularities

The last two weeks will be devoted to geometric optics approximation and propagation of disturbances following Chapters 4-5 in Rauch Hyperbolic Partial differential Equations and Geometric Optics Next time I will try to include Hilbert space methods to find a solution.