Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)

Maurice De Gosson

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 3, page 183-213
  • ISSN: 0373-0956

Abstract

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This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted ∂ W F ω ( u ) for distributions u ∈ D ' ( R + n ) , regular in the normal variable x n (thus, ∂ W F ω ( u ) = ∅ means that u ∈ ∩ s + t = 1 / 2 H s + t near the boundary), and it is shown that ∂ W F ω - m [ P ( u 0 ) x n > 0 ] is a subset of ∂ W F ( u ) if P has degree m and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential Cauchy problems, with bicharacteristics transversal to the hyperplane supporting the Cauchy data.

How to cite

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De Gosson, Maurice. "Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)." Annales de l'institut Fourier 32.3 (1982): 183-213. <http://eudml.org/doc/74545>.

@article{DeGosson1982,
abstract = {This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted $\partial WF_\omega (u)$ for distributions $u \in D^\{\prime \}(\{\bf R\}^n_+)$, regular in the normal variable $x_n$ (thus, $\partial WF_\omega (u)= \emptyset$ means that $u\in \cap _\{s+t=1/2\} H^\{s+t\}$ near the boundary), and it is shown that $\partial WF_\{\omega -m\} [P(u^0)_\{x_n&gt;0\}]$ is a subset of $\partial WF(u)$ if $P$ has degree $m$ and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential Cauchy problems, with bicharacteristics transversal to the hyperplane supporting the Cauchy data.},
author = {De Gosson, Maurice},
journal = {Annales de l'institut Fourier},
keywords = {boundary singular spectrum; differential Cauchy problems; bicharacteristic transversal},
language = {eng},
number = {3},
pages = {183-213},
publisher = {Association des Annales de l'Institut Fourier},
title = {Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)},
url = {http://eudml.org/doc/74545},
volume = {32},
year = {1982},
}

TY - JOUR
AU - De Gosson, Maurice
TI - Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 3
SP - 183
EP - 213
AB - This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted $\partial WF_\omega (u)$ for distributions $u \in D^{\prime }({\bf R}^n_+)$, regular in the normal variable $x_n$ (thus, $\partial WF_\omega (u)= \emptyset$ means that $u\in \cap _{s+t=1/2} H^{s+t}$ near the boundary), and it is shown that $\partial WF_{\omega -m} [P(u^0)_{x_n&gt;0}]$ is a subset of $\partial WF(u)$ if $P$ has degree $m$ and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential Cauchy problems, with bicharacteristics transversal to the hyperplane supporting the Cauchy data.
LA - eng
KW - boundary singular spectrum; differential Cauchy problems; bicharacteristic transversal
UR - http://eudml.org/doc/74545
ER -

References

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  6. [6] M. DE GOSSON, Résultats microlocaux en hypoellipticité partielle à la frontière pour les O.P.D. de transmission, C.R. Acad Sc., Paris, t. 292. Zbl0469.35080
  7. [7] L. HÖRMANDER, Linear partial differential operators. Springer Verlag, 1964. Zbl0108.09301
  8. [8] L. HÖRMANDER, Pseudo-differential operators and non-elliptic boundary problems. Ann. Math., 83 (1966). Zbl0132.07402MR38 #1387
  9. [9] L. HÖRMANDER, On the existence and the regularity of solutions of linear pseudo-differential equations, L'Ens. Math., t. XVII, fasc. 2 (1972). 
  10. [10] LIONS, MAGENES, Problèmes aux limites non homogènes et applications, vol. I, Dunod, 1968. Zbl0165.10801
  11. [11] R.B. MELROSE, Transformations of boundary problems, Preprint, Acta Math. (1980). Zbl0436.58024
  12. [12] R.B. MELROSE, J. SJÖSTRAND, Singularities of boundary problems, I, Comm. on Pure and Appl. Math., XXXI (1978). Zbl0368.35020
  13. [13] F. TREVES, Linear partial differential equations with constant coefficients, Gordon and breach, 1963. 
  14. [14] J. SJÖSTRAND, Operators of principal type with interior boundary conditions, Acta Math., 130 (1973). Zbl0253.35076MR55 #9174

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