On the smoothness of solution of the mixed boundary value problem for the second order hyperbolic equation in a neighbourhood of an edge (Q1175291)

The author considers the second order hyperbolic equation \[ {\mathfrak L}u\equiv u_{tt}-Lu=f(x,t),\quad Lu=\sum^ n_{i,j=1}{\partial\over\partial x_ i}(a_{ij}(x,t)u_{x_ j})+\sum^{n+1}_{i=1}a_ i(x,t)u_{x_ i}+a(x,t)u, \leqno (1) \] \(x_{n+1}=t\), \(a_{ij}=a_{ji}\), \(\nu\xi^ 2\leq a_{ij}\xi_ i\xi_ j\leq\mu\xi^ 2\), \(\nu>0\), where \(a_{ij}(x,t)\), \(a_ i(x,t)\), \(a(x,t)\) are real functions having infinite smoothness in the cylinder \(\overline Q_ T=\overline G\times[0,T]\) where \(\overline G\) is the closure of a given bounded domain \(G\) whose boundary is piecewise smooth, contained in (\(n-1\))-dimensional smooth surfaces \(\Gamma_ i (i=1,2,\ldots,m)\). Suppose that the surface \(\Gamma_ i\) can intersect only \(\Gamma_{i-1}\), \(\Gamma_{i+1}\) along \((n-2)\)-dimensional smooth manifolds \(\ell_{i-1}\), \(\ell_{i+1}\), respectively. The mixed boundary value problem for equation (1) consists of the following initial and boundary conditions (2) \(u|_{t=0}=\varphi(x)\), \(u_{t| t=0}=\psi(x)\), \(u|_{S_ 1}=0\), \({\partial\over \partial N}_{\mid S_ 2}=0\), where \(S_ i=\Gamma_ i\times [0,T] (i=1,2)\). Under suitable regularity and boundedness conditions on \(L\), \(f\) and the data, problem (1)--(2) has a weak solution in \(W^ 1(Q_ T)\) such that \[ | u|_{W^ 1(Q_ T)}\leq c(T)(|\varphi|_{W^ 1(G)}+|\psi|_{L_ 2(G)}+| f|_{L_{2,1}(Q_ T)}) \] where \(c(T)=const>0\) and \(c(T)\) does not depend on the functions \(u\), \(\varphi\), \(\psi\) and \(f\). A regularity theorem is also proved under further smoothness hypotheses on \(L\) and \(f\) with zero initial data.