On higher-codimensional boundary value problems

Abstract

Let us consider a real analytic manifold M and its submanifold N of codimen-sion D. We take Y⊂ X as a complexification of N ⊂ M. Let M be a coherent module over the sheaf of ring D _x of holomorphic differential operators on X and assume that Y is noncharacteristic for M. In this talk, we first report that we have weakened the conditions on M for the vanishing of the local cohomologies of the hyperfunction solution complex :

$${H^j}\mu NRHom{\varepsilon _{Dx}}(M,{B_M}) \simeq 0\;for\;j < d,$$

(1)

where μ _N denotes Sato’s microlocalization functor along N. This kind of problem is very important since many classical results in analysis can be deduced from such vanishing theorems on cohomologies. Holmgren’s uniqueness theorem and the famous abstract Edge of the Wedge theorem of Martineau [14], Morimoto [15] and Kashiwara [7] (when M=∂˜ the Cauchy-Riemann equation) are special cases of it. In particular, the last result was essentially used to construct the microfunction theory by Sato et al. [17] and many extension theorems on holomorphic functions can be obtained from it. For example, we have the Edge of the Wedge theorem of Martineau for holomorphic functions and the local Bochner’s tube theorem of Komatsu [13]. After that, Kashiwara-Kawai [6] showed (1) under the additional condition of the ellipticity of M and extended this result to general systems. In this talk, we shall further extend it into various cases where the solutions are not necessarily real analytic, and we derive several results on the extension of hyperfunction solutions. For this purpose, we make use of the theory of bimicrolocalization developped in [21] and [24] showing in particular that the sheaf RHom _Dx (M, C _nm ) of bimicrofunction solutions of [21] plays the role of the obstruction against the extension of hyperfunction solutions. We also prove (under weaker conditions than that of Kashiwara-Kawai [6]) that every real analytic solutions on an open convex cone Ω ⊂ M = IRⁿ with the edge N automatically extends to a conic open neighborhood of Ω in X = ℂⁿ. To investigate this phenomenon, a partial solution to Schapira’s conjecture and the result on the injectivity of the microlocal boundary value morphism proved in [23] will be used.