Well posedness of hyperbolic Initial Boundary Value Problems

Well-posedness of hyperbolic Initial Boundary Value Problems Jean-Franc¸ois Coulombel CNRS & Universite Lille 1 Laboratoire de mathematiques Paul Painleve Cite scientifique 59655 VILLENEUVE D'ASCQ CEDEX, France e-mail: March 16, 2004 Abstract Assuming that a hyperbolic initial boundary value problem satsifies an a priori energy estimate with a loss of one tangential derivative, we show a well-posedness result in the sense of Hadamard. The coefficients are assumed to have only finite smoothness in view of applications to nonlinear problems. This shows that the weak Lopatinskii condition is roughly sufficient to ensure well-posedness in appropriate functional spaces. AMS subject classification: 35L50, 35L40 1 Introduction In this paper, we consider hyperbolic Initial Boundary Value Problems (IBVPs) in several space dimensions. Such problems typically read: ? ?? ?? ∂tU + ∑d j=1Aj(t, x) ∂xjU +D(t, x)U = f(t, x) , t ? ]0, T [ , x ? R d + , B(t, y)U|xd=0 = g(t, y) , t ? ]0, T [ , y ? R d?1 , U|t=0 = U0(x) , x ? R d + .

• space rd?


• shall call semi-strong


• problems usually


• lopatinskii condition


• lopatinskii condition yields