Higher-Codimensional Boundary Value Problems and $F$-Mild Microfunctions —Local and Microlocal Uniqueness—

For the study of local and microlocal boundary value problems with a boundary of codimension greater than one, sheaves of F-mild hyperfunctions and F-mild microfunctions are introduced. They are refinements of the notions of hyperfunctions and microfunctions with real analytic parameters and have natural boundary values. F-mild solutions of a general ^-Module M (that is, a system of linear partial differential equations with analytic coefficients) are considered. In particular, local and microlocal uniqueness in the boundary value problem (the Holmgren type theorem) is proved if the boundary is non-characteristic for M. or else if M is Fuchsian along the boundary. Introduction The purpose of this paper is to study the higher-codimensional boundary value problem for a general system of linear partial differential equations with analytic coefficients. In general, we must impose some regularity condition on the solutions in order to define their boundary values. We introduce the notion of F-mild hyperfunctions as this regularity condition, which is a refinement of that of hyperfunctions with real analytic parameters. We also define the notion of F-mild microfunctions as a microlocalization of that of F-mildness. Our main result is the local and microlocal uniqueness of F-mild hyperfunction (or microfunction) solutions of a system of linear partial differential equations which is Fuchsian along Y in the sense of Y. Laurent and T. Monteiro Femandes [L-MF] or in the sense of N. S. Madi [M] and S. Yamazaki [Y]. Communicated by T. Kawai, March 20, 1998. Revised June 22, 1998. 1991 Mathematics Subject ClassificationPrimary 35G15; Secondary 32A45. 35A27. Department of Mathematical Sciences, Yokohama City University. 22-2 Seto, Kanazawa-ku, Yokohama 236-0027 Japan. Research Fellow of The Japan Society for The Promotion of Science, Graduate School of Mathematical Sciences, The University of Tokyo, 8-1 Komaba 3-chome, Meguro-ku, Tokyo 153-8914,Japan 384 F-MlLD MlCROFUNCTIONS Let M be a real analytic manifold and N a closed real analytic submanifold of M of codimension d> 2, Then the sheaf $N\M of F-mild hyperfunctions is defined on the normal bundle T#M of N (strictly speaking, the sheaf S&NIM depends on a partial complexification L of M). Let us take a local coordinate system (t, x) = (&,..., fe, xi,..., x«) of M such that N is defined by £—0. Assume that / is a section of S^NIM (that is, an F-mild hyperfunction) defined on a neighborhood of 0 + 9/9/i^TVM Then /is actually regarded as a hyperfunction defined on a wedge domain with edge A for some £>0. In addition, for any non-negative integers «!,...,#*, goi ... Qfdd f(^ x) kas a natural boundary value as £ tends to zero as a hyperfunction of x. The restriction of S$N\M to the zero-section of T#M coincides with the sheaf of hyperfunctions defined on a neighborhood of N which have t as real analytic parameters. Moreover, a section of %N\M which is defined on T#M with the zerosection removed has also t as real analytic parameters on a neighborhood of N. Hence we may regard $N\M as a tangential decomposition of the sheaf of hyperfunctions which have t as real analytic parameters. We take complexifications X and Y of M and N respectively such that Y is a closed submanifold of X. We denote by $* the sheaf on X of rings of linear partial differential operators (of finite order) with holomorphic coefficients. Let M be a coherent left ©^-Module; that is, a system of linear partial differential equations with holomorphic coefficients (in this paper, we shall write Module with a capital letter, instead of sheaf of modules) . First, let us assume that Y is non-characteristic for M. Then we prove that any hyperfunction solution to M which is defined on a wedge domain with edge N is F-mild, thus having boundary values with no further assumption. This case was studied by P. Schapira ( [Sc 1] , [Sc 2] ) by using the theory of microlocalization of sheaves. The local uniqueness in this boundary value problem was proved in T. Oaku [04]. K. Takeuchi [Tk] formulated microlocal boundary value problem by using the theory of second microlocalization and proved the microlocal uniqueness in the non-characteristic case. Here we give another proof to the microlocal uniqueness by a natural extension of the method used in Oaku [04]. Next, suppose that M is Fuchsian along Y in the sense of Laurent and Monteiro Fernandes [L-MF] . In this case, not all the hyperfunction solutions to M are necessarily F-mild, but we can obtain the local and microlocal uniqueness for F-mild solutions. More precisely, we obtain a monomorphism (an injective sheaf homomorphism) TOSHINORI OAKU AND SUSUMU YAMAZAKI 385 where TN: TNM—+N is the canonical projection, $x is the sheaf of hyperfunctions on N, and My is the induced system (that is, the ©-Module theoretic restriction) of M to Y, which is a coherent ©y-Module. We can also obtain the microlocalization of this morphism, which is also injective. Finally assume that M is a Fuchs-Goursat system in the sense of Yamazaki [Y] , which is a generalization of a Fuchs-Goursat operator due to Madi [M] . In this case, since My is not coherent over ©r in general, we consider a kind of Goursat problem: Set Mi={(t, x) ^^ xR: /, = ()} by using a local coordinate system as mentioned above. For an F-mild hyperfunction, we can define its restriction to Mt for Ki0 (Ki 1} and No = NU{0}. Let M be a (d -r n) -dimensional real analytic manifold and N a ndimensional closed real analytic submanifold of M. In this paper, we always assume that d^ 2. There exist complexifications X and Y of M and N respectively such that Y is a closed submanifold of X. We assume that there exists a (d + 2n) -dimensional real analytic submanifold L of X containing both M and Y such that the triplet (N, M, L) is locally isomorphic to the triplet ({0} xE, M+, W x(C) by a local coordinate system (r, *) of X around each point of N. We say such a local coordinate system admissible. We use the notation T=t+^ls(^ s^m r ) , z^x+J^ly (x, y^E) , \z =max{Uft ; Kfe <7i} and so on for an admissible local coordinate system (r, z). Hence by an admissible local coordinate system the following inclusion relations are obtained: r = l o l We shall mainly follow the notation of Kashiwara-Schapira [K-S 2]; we denote the normal deformations of N and Y in M and L by MN and LK respectively. For example, by an admissible coordinate system, we see that MN— {(r, t, x): r^H, (r, f, x) eM), ^M^M^n {(r, t x); r>0}, TNM~MNn {(r, t .r); r^O} and pM: MJV ^ (r, fc x) (r, f, xj ^ M. Then, we can regard MN as a submanifold of Ly. Therefore we have the following commutative diagram: TOSHINORI OAKU AND SUSUMU YAMAZAKI 387