A sharp symmetrization of the \(L^ p\)-well-posed mixed problem for regularly hyperbolic equations of second order
The paper is a continuation of a previous article of the author and \textit{F. Suzuki} [Jap. J. Math. 20, 133-186 (1994; Zbl 0809.35051)]. It concerns the equation \[ \begin{multlined} {\partial^2u\over\partial t^2}- 2\sum^n_{j=1} h_j(t,x) {\partial ^2u\over\partial t\partial x_j}- \sum^n_{i,j= 1}a_{ij}(t, x){\partial^2u\over\partial x_i\partial x_j}+ a_0(t,x){\partial u\over\partial t}+\\ \sum^n_{j=1} a_j(t,x){\partial u\over\partial x_j}+ e_0(t, x)u= f(t,x)\end{multlined} \] with initial conditions \(u(0,x)= u_0(x)\), \(u_t(0, x)= u_1(x)\) and boundary conditions \[ \begin{multlined} a_{11}(t,0, x')^{-1/2}\Biggl\{a_{11}(t, 0,x'){\partial u\over\partial x_1}+ \sum^n_{j= 2} a_{1j}(t, 0, x'){\partial u\over\partial x_j}+ h_1(t,0, x'){\partial u\over\partial t}\Biggr\}+\\ \sum^n_{j= 2}b_j(t, x'){\partial u\over\partial x_j}- c(t,x')\Biggl(1+ {h_1(t, 0,x')^2\over a_{11}(t, 0, x')}\Biggr)^{1/2}\cdot\Biggl\{{\partial u\over\partial t}-\\ \Biggl(1+ {h_1(t,0, x')^2\over a_{11}(t, 0, x')}\Biggr)^{-1} \sum^n_{j= 2} \Biggl(h_j(t, 0,x')- {h_1(t,0, x')\over a_{11}(t, 0, x')} a_{1j}(t, 0,x')\Biggr){\partial u\over\partial x_j}\Biggr\}+\\ \gamma(t, x')u|_{x_1= 0}= g(t,x'),\end{multlined} \] where \((t,x)= (t, x_1,x')\in \mathbb{R}^1_+\times \mathbb{R}^1_+\times \mathbb{R}^{n-1}\). The author obtains, under certain conditions, a complete and sharp symmetrization.
scientific article; zbMATH DE number 879999
scientific article