Initial-boundary value problems for second order systems of partial differential equations∗
Träsko-StoröInstitute of Mathematics,
NADA, KTH, 100 44
2 Facultad de Matemáticas, Astronomía y Física and IFEG, Universidad Nacional de Córdoba, Ciudad Universitaria, CP :X5000HUA, Córdoba, Argentina
3 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, California, USA
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order system. Here we will use the theory of pseudo-differential operators combined with mode analysis. There are many desirable properties of this approach: (1) the reduction to first order systems of pseudo-differential equations poses no difficulty and always gives a system of 2n equations. (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.
Mathematics Subject Classification: 35L20 / 65M30
Key words: Well-posed 2nd-order hyperbolic equations / surface waves / glancing waves / elastic wave equation / Maxwell equations
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. O.E.O. acknowledges support by grants 05/B415 and 214/10 from SeCyT-Universidad Nacional de Córdoba, 11220080100754 from CONICET, PICT17-25971 from ANPCYT, and the Partner Group grant of the Max Planck Institute for Gravitational Physics, Albert-Einstein-Institute (Germany).
© EDP Sciences, SMAI, 2012