Persistence of regularity of the solution to a hyperbolic boundary value problem in domain with corner

Abstract

This article is about the question of the persistence of regularity for the solution to hyperbolic boundary value problem in the quarter-space. More precisely we will both consider the pure boundary value problem and the initial boundary value problem and we propose a functional space, based upon the high order Sobolev space in which a control of the data of the problem leads to a control of the solution (in the same space). The space proposed here contains the tangential Sobolev space. The analysis borrows some ideas of the study of characteristic boundary value problems in the half-space for which the good derivative to consider is known to be the tangential derivatives x1∂1 instead of the normal derivative ∂1. For quarter-space problems the good quantity to consider will be the radial derivative x1∂1+x2∂2 and then we recover the control of tangential derivatives x1∂1 and x2∂2 using explicit formulas in polar coordinates. The regularity of the solution is then established intrinsically by adapting the method introduced by the author to deal with half-space problems without using regularization methods.