New Trends in Microlocal Analysis pp 225-234 | Cite as

# On higher-codimensional boundary value problems

Chapter

## Abstract

Let us consider a real analytic manifold where μ

*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)

_{ 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*= IR^{ n }with the edge*N*automatically extends to a conic open neighborhood of Ω in*X*= ℂ^{ n }. 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.## Keywords

Elliptic System Local Cohomology Microlocal Analysis Real Analytic Manifold Closed Submanifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [1]T. Aoki and S. Tajima,
*On a generalization of Bochner’s tube theorem for C-R-submanifolds*, Proc. Japan Acad., Ser. A,**63**(1987), 302–303.MathSciNetMATHCrossRefGoogle Scholar - [2]J.M. Bony and P. Schapira,
*Propagation des singularités analytiques pour les solutions des équations aux derivées partielles*, Ann. Inst. Fourier, Grenoble,**26**(1976), 81–140.MathSciNetMATHCrossRefGoogle Scholar - [3]J-M. Delort,
*Microlocalisation simultanée et problème de Cauchy ramifié*, to appear in Compositio Math.Google Scholar - [4]S. Funakoshi, Master thesis presented to the University of Tokyo, 1995.Google Scholar
- [5]A. Kaneko,
*Singular spectrum of boundary values of solutions of partial differential equations with real analytic coefficients*, Sci. Papers College Gen. Ed. Univ. Tokyo, 25 (1975), 59–68.MATHGoogle Scholar - [6]M. Kashiwara and T. Kawai,
*On the boundary value problem for elliptic systems of linear partial differential equations I–II*, Proc. Japan Acad.,**48**(1971), 712–715; ibid.,**49**(1972), 164-168.MathSciNetCrossRefGoogle Scholar - [7]M. Kashiwara and Y. Laurent,
*Théorèmes d’annulation et deuxième microlocalisation*, prépublication d’Orsay, 1983.Google Scholar - [8]M. Kashiwara and P. Schapira,
*Micro-hyperbolic systems*, Acta Math.**142**(1979), 1–55.MathSciNetMATHCrossRefGoogle Scholar - [9]M. Kashiwara and P. Schapira,
*Sheaves on manifolds*Grundlehlen der Math. Wiss.**292**, Springer-Verlag, 1990.Google Scholar - [10]K. Kataoka,
*Microlocal theory of boundary value problems I–II*, J. Fac. Sci. Univ. Tokyo**27**(1980), 355–399; ibid.,**28**(1981), 31-56.MathSciNetGoogle Scholar - [11]K. Kataoka and N. Tose,
*Some remarks in 2nd microlocalization*(in Japanese), RIMS Kokyuroku, Kyoto Univ.**660**(1988), 52–63.Google Scholar - [12]T. Kawai,
*Extension of solutions of systems of linear differential equations*, Publ. RIMS, Kyoto Univ.**12**(1976), 215–227.MATHCrossRefGoogle Scholar - [13]H. Komatsu,
*A local version of Bochner’s tube theorem*, J. Fac. Sci. Univ. Tokyo**19**(1972), 201–214.MathSciNetMATHGoogle Scholar - [14]A. Martineau,
*Le ‘edge of the wedge theorem’ en théorie des hyperfonctions de Sato*, Proc. Intern. Conf. on Functional Analysis and Related Topics, 1969, Univ. Tokyo Press, Tokyo, 1970, pp. 95–106.Google Scholar - [15]M. Morimoto,
*Sur la décomposition du faisceau des germes de singularités d’hyperfonctions*, J. Fac. Sci. Univ. Tokyo**17**(1970), 215–239.MathSciNetMATHGoogle Scholar - [16]T. Oaku,
*Higher-codimensional boundary value problem and F-mild hyperfunctions*, in Algebraic Analysis Vol.**II**, (Papers dedicated to Prof. Sato), (M. Kashiwara and T. Kawai (eds.)), Academic Press, 1988, pp. 571–586.Google Scholar - [17]M. Sato, T. Kawai and M. Kashiwara,
*Hyperfunctions and pseudodifferential equations*Lecure Notes in Math.**287**, Springer-Verlag, 1973, pp. 265–529.MathSciNetGoogle Scholar - [18]P. Schapira,
*Propagation at the Boundary of Analytic singularities*, Singularities of Boundary Value Problems, Reidel Publ. Co., 1981, pp. 185–212.Google Scholar - [19]P. Schapira,
*Front d’onde analytique au bord II*, Séminaire E.D.P., École Polyt., 1986, Exp.13.Google Scholar - [20]P. Schapira,
*Microfunctions for boundary value problems*, Algebraic Analysis (Papers dedicated to Prof. Sato) ( M. Kashiwara and T. Kawai (eds.)), Academic Press, 1088, pp.809–819.Google Scholar - [21]P. Schapira and K. Takeuchi,
*Déformation binormale et bispécialisation*, C.R. Acad. Sc.**319**, Série I (1994), 707–712.MathSciNetMATHGoogle Scholar - [22]P. Schapira and G. Zampieri,
*Regularity at the boundary for systems of microdifferential operators*, Pitman Research Notes in Math.158, 1987, pp. 186–201.Google Scholar - [23]K. Takeuchi,
*Microlocal boundary value problems in higher codimensions*, to appear in Bull. Soc. math. France, t.**124**(1996), 243–276.MATHGoogle Scholar - [24]K. Takeuchi,
*Binormal deformation and bimicrolocalization*, Publ. RIMS, Kyoto Univ.,**32**(1996), 115–160.CrossRefGoogle Scholar - [25]K. Takeuchi,
*Théorèmes de type Edge of the Wedge pour les solutions hyperfonctions*, C.R. Acad. Sc.**321**, Série I (1995), 1333–1336.MATHGoogle Scholar - [26]K. Takeuchi,
*Edge of the Wedge type theorems for hyperfunction solutions*, submitting.Google Scholar - [27]N. Tose,
*Theory of partially elliptic systems and its applications*, Master thesis presented to the University of Tokyo, 1985.Google Scholar - [28]M. Uchida,
*Continuation of analytic solutions of linear differential equations up to convex conical singularities*, Bull. Soc. math. France, t.**121**(1993), 133–152.MathSciNetMATHGoogle Scholar - [29]M. Uchida
*A generalization of Bochner’s tube theorem for elliptic boundary value problems*, RIMS Kokyuroku, Kyoto Univ.**845**(1993), 129–138.MathSciNetGoogle Scholar

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