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.

Classification: 35L20, 65M30

Keywords: well-posed 2nd-order hyperbolic equations, surface waves, glancing waves, elastic wave equation, Maxwell equations

@article{M2AN_2012__46_3_559_0, author = {Kreiss, Heinz-Otto and Ortiz, Omar E. and Anders Petersson, N.}, title = {Initial-boundary value problems for second order systems of partial differential equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {46}, number = {3}, year = {2012}, pages = {559-593}, doi = {10.1051/m2an/2011060}, zbl = {1276.35115}, mrnumber = {2877365}, language = {en}, url = {http://www.numdam.org/item/M2AN_2012__46_3_559_0} }

Kreiss, Heinz-Otto; Ortiz, Omar E.; Anders Petersson, N. Initial-boundary value problems for second order systems of partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 559-593. doi : 10.1051/m2an/2011060. http://www.numdam.org/item/M2AN_2012__46_3_559_0/

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