Non homogeneous boundary value problems for second order hyperbolic operators (Q1094586)

The authors deal with the regularity problem over a finite interval [0,T], \(T<+\infty\), for the second order hyperbolic mixed problem with Dirichlet boundary conditions, namely \[ u_{tt}+A(x,t)u=f\quad in\quad Q=\Omega \times (0,T) \] \[ u(x,0)=u^ 0(x),\quad u_ t(x,0)=u^ 1(x)\quad in\quad \Omega;\quad u=g\quad on\quad \Sigma =\partial \Omega \times (0,T), \] where \(\Omega\) is an open bounded domain of \({\mathbb{R}}^ n\) with boundary assumed to be as smooth as necessary and A denotes a second order elliptic operator in \(\Omega\), that is \[ A(x,t)=- \sum^{n}_{i,j=1}\frac{\partial}{\partial x_ i}(a_{ij}(x,t)\frac{\partial}{\partial x_ j});\quad a_{ij}=a_{ji}\in L^{\infty}(Q),\quad i,j=1,...,n;\quad \sum^{n}_{i,j=1}a_{ij}(x,t)\xi_ i\xi_ j\geq \gamma \sum^{n}_{i=1}\xi^ 2_ i,\quad \gamma >0. \] Under various assumptions of regularity of f in Q, g in \(\Sigma\), \(u^ 0\), \(u^ 1\) on \(\Omega\), and, if necessary, appropriate compatibility conditions, the authors discuss the optimal interior regularity of u, \(u_ t\), etc., in Q as well as the boundary regularity of \(u_{\nu}\) on \(\Sigma\), where \(\nu\) is the unit outward normal to \(\partial \Omega\). The main purpose of this paper is to offer a comprehensive treatment and a simpler presentation of the regularity problem under consideration which, in particular, includes the previous results of the authors for g in \(L^ 2(\Sigma)\) and of \textit{R. Sakamoto} [J. Math. Kyoto Univ. 10, 349-373 (1970; Zbl 0203.100) and ibid., 403-417 (1970; Zbl 0206.401)] for g in \(H^ k(\Sigma)\), \(k\geq 1\), at least for second order problems and bounded domains \(\Omega\). All the results here established are first proved in the canonical case where \(A(x,t)=-\Delta\) by means of a direct method and a functional analytic method based on representation formulas. The direct method is then applied to the general case when \(A=A(x,t)\); while the representation formulas are extended to the case in which \(A=A(x)\) does not depend on t.