On the mixed problem for wave equation in a domain with a corner
The author generalizes the results from his paper [Tokyo J. Math. 5, 183- 211 (1982; Zbl 0529.35050)]. For example he considers the problem (1) \[ L_ 1[u]=\frac{\partial^ 2u}{\partial t^ 2}-\frac{\partial^ 2u}{\partial x^ 2}-\frac{\partial^ 2u}{\partial y^ 2}+du=f(t,x,y);\quad u(0,x,y)=u_ 0(x,y),\quad u_ t(0,x,y)=u_ 1(x,y) \] \[ B_ 1[u]|_{x=0}=(\partial u/\partial x+b\partial u/\partial y-c\partial u/\partial t+\alpha u)|_{x=0}=g_ 1(t,y) \] \[ B_ 2[u]|_{y=0}=(\partial u/\partial y+(1/b)\partial u/\partial x- (c/b)\partial u/\partial t+(\alpha /b)u)|_{y=0}=g_ 2(t,x). \] In his paper [loc. cit.] he showed that the problem (1) is \(L^ 2\)-well- posed if the mixed problems (2) \[ L_ 1[u]=f(t,x,y);\quad u(0,x,y)=u_ 0(x,y),\quad u_ t(0,x,y)=u_ 1(x,y);\quad B_ 1[u]|_{x=0}=g_ 1(t,y) \] and (3) \[ L_ 1[u]=f(t,x,y);\quad u(0,x,y)=u_ 0(x,y),\quad u_ t(0,x,y)=u_ 1(x,y);\quad B_ 2[u]|_{y=0}=g_ 2(t,x) \] are \(L^ 2\)-well-posed and \(b\neq i\). In this paper he shows that the problem (1) is \(L^ 2\)-well-posed if the problems (2) and (3) are \(L^ 2\)-well- posed and \(b=\pm i\). Two similar mixed problems in a domain with a corner are considered and energy inequalities are obtained by reducing the wave equation to a symmetric hyperbolic systems of first order.
symmetric hyperbolic systems of first order
scientific article; zbMATH DE number 3989942