## UC Berkeley / Lawrence Berkeley Laboratory

#### The initial boundary value problems for symmetric second order hyperbolic systems

**Heinz Kreiss, Swedish Academy of Sciences**

##### November 2nd, 2011 at 2PM–3PM in 891 Evans Hall [Map]

Recently, Anders Petersson, Omar Ortiz and myself have developed a rather
general theory for second order hyperbolic systems based on Laplace and Fourier
transforms, with particular emphasis on boundary processes corresponding to
generalized eigenvalues. Our theory uses mode analysis and builds upon the
theory for first order systems, developed a long time ago. This approach has
many desirable properties: 1) Once a second order system has been Laplace and
Fourier transformed it can always be written as a system of 2*n* first
order pseudo-differential equations. Therefore the above theory also applies
here. 2) We can localize the problem, *i.e.*, it is only necessary to
study the Cauchy problem and halfplane problems with constant coefficients. 3)
The class of problems we can treat is much larger than previous approaches
based on “integration by parts”. 4) The relation between boundary
conditions and boundary phenomena becomes transparent.

I will not bore you with complicated proofs. Instead, I will discuss a number of examples.

- Wave equation (Kreiss, Ortiz, Petersson)
- Einstein's equation of general relativity (harmonic gage) (Kreiss, Winicour)
- Elastic wave equation (Kreiss, Petersson)
- Characteristic boundary conditions (Kreiss, Winicour).