On the micro-hyperbolic boundary value problem for systems of differential equations (Q5935924)

Let \(M\) be a real analytic manifold, \(N\) a submanifold of \(M\) of codimension 1 defined by the equation \(f=0,\) where \(f\) is a real-valued analytic function with \(df|_N\not=0.\) Let \(Z_+\) be the closed domain in \(M\) with analytic boundary defined by the condition \(f\geq 0.\) In [Proc. Japan Acad. 48, 712-715 (1972; Zbl 0271.35028) and Proc. Japan Acad. 49, 164-168 (1973; Zbl 0279.35037)], \textit{M. Kashiwara} and \textit{T. Kawai} described the cohomology groups of \(R\Gamma_{Z_+}R\Hom_{{\mathcal D}_M}({\mathcal M},{\mathcal B}_M)|_N,\) where \({\mathcal M}\) is an elliptic \({\mathcal D}_M\)-module, in terms of a system of microdifferential equations induced on the boundary. NEWLINENEWLINENEWLINEIn this paper, the author extends the Kashiwara-Kawai formula to the case of a coherent \({\mathcal D}_M\)-module \({\mathcal M}\) which satisfy the following condition of micro-hyperbolicity at the boundary NEWLINE\[NEWLINEdf(p)\notin\text{ Witney normal cone }(\text{Char}({\mathcal M}), Z_+\times_M T^*_MX), NEWLINE\]NEWLINE NEWLINE\[NEWLINE\forall p\in\{(x,\xi)\in T^*_MX;\;x\in N,\;\xi\not=0\},NEWLINE\]NEWLINE where \(X\) is a complex neighborhood of \(M.\) The author is also able to define, upon denoting by \(Y\) a codimension 1 complex neighborhood of \(N\) and assuming that the embedding \(Y\rightarrow X\) is non-characteristic for \({\mathcal M}\), a coherent \({\mathcal E}_Y\)-module \({\mathcal N}^+\) and proves that NEWLINE\[NEWLINER\Gamma_{Z_+}R\Hom_{{\mathcal D}_X}({\mathcal M},{\mathcal B}_M)|_N[1]\simeq R\dot{\pi}_{N*}R\Hom_{{\mathcal E}_Y}({\mathcal N}^+,{\mathcal C}_N),NEWLINE\]NEWLINE a formula that can be microlocalized by using the sheaf \({\mathcal C}_{Z_+|X}\) of Kataoka. Hence the author is able to reformulate in the derived category of sheaves for systems of differential equations the main parts of work of \textit{A. Kaneko} [Sci. Pap. Coll. Gen. Educ., Univ. Tokyo 25, 59-68 (1975; Zbl 0332.35048)] and \textit{T. Oaku} [Proc. Japan Acad., Ser. A 55, 136-140 (1979; Zbl 0434.35005)] (Section 3).