Boundary value problems for systems with a parameter

M. S. Agranovich

Sbornik. Mathematics, Tome 13 (1971) no. 1, pp. 25-64

Résumé

The paper is concerned with problems of the form $$ A\biggl (x,\frac\partial{\partial x},p\biggr)u(x)=f(x)\quad\text{in}\quad G,\qquad B\biggl(x,\frac\partial{\partial x},p\biggr)u(x)=g(x)\quad\text{on}\quad\Gamma. $$ Here $G$ is a region in $R_x^n$ with smooth boundary $\Gamma$; $A$ and $B$ are matrices of linear partial differential operators with smooth coefficients, depending polynomially on the complex parameter $p$. The operator $A$ is obtained by replacing $\partial/\partial x$ by $p$ in the operator $A(x,\partial/\partial x,\partial/\partial t)$, which is strongly hyperbolic in the sense of I. G. Petrovskii. Under some supplementary assumptions, the existence and uniqueness of a strong solution in the spaces $H_{qs}$ is demonstrated, and an a priori estimate in norms involving the parameter $p$ is obtained for large values $\operatorname{Re}p$. Bibliography: 30 titles.

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