Loss of derivatives for hyperbolic boundary problems with constant coefficients

Abstract

Symmetric hyperbolic systems and constantly hyperbolic systems with constant coefficients and a boundary condition which satisfies a weakened form of the Kreiss-Sakamoto condition are considered. A well-posedness result is established which generalizes a theorem by Chazarain and Piriou for scalar strictly hyperbolic equations and non-characteristic boundaries [3]. The proof is based on an explicit solution of the boundary problem by means of the Fourier-Laplace transform. As long as the operator is symmetric, the boundary is allowed to be characteristic.

Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy

“…However if such a weak well-posedness theory is available then we can show (see Section 4) that the solution to the geometric optics expansion is effectively an approximate solution. Moreover for some examples like the interesting case of the wave equation for Neumann boundary condition such a weak well-posedness theory exists (see [11] and [10]) so, on some examples, we can also conclude that the geometric optics expansion is an approximate solution.…”

“…But let us say that in some particular setting such estimates can be found in the litterature (see [11] and [10]). More precisely in [11] the author obtains a weak well-posedness result when the so-called Kreiss-Sakamoto condition with power s holds.…”

“…But let us say that in some particular setting such estimates can be found in the litterature (see [11] and [10]). More precisely in [11] the author obtains a weak well-posedness result when the so-called Kreiss-Sakamoto condition with power s holds. Even if a full characterization of the fulfilment of the Kreiss-Sakamoto condition in terms of the area of degeneracy of the uniform Kreiss-Loaptinskii condition as not achieved yet we can use this estimate for the very interesting (in view of the applications) 2d wave equation with Neumann boundary condition.…”

New notions and constructions of the boundary control method

“…The space of controls F T contains the nest [43]. However, it is well defined on a class M T of smooth controls vanishing near t = 0.…”

“…By the isometry C op (Ω) ↔ C(Ω), the relation ( 44) is equivalent to E = C op (Ω), whereas (43) leads to the relation…”