Uniqueness of solutions of partial differential equations when the initial surface is characteristic at a point

Korbly, Letitia J.

doi:10.1090/S0002-9939-1982-0674093-X

Uniqueness of solutions of partial differential equations when the initial surface is characteristic at a point
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Proc. Amer. Math. Soc. 86 (1982), 617-624 Request permission

Abstract:

Uniqueness in the Cauchy problem for hyperbolic operators degenerate at a point on the initial surface depends on values of the coefficients of the lower order terms. If the operator $P$ is doubly characteristic at the origin with respect to the $t = 0$ line, $P$ has uniqueness for functions which are smooth enough if the coefficient of the ${D_t}$ term does not lie in a certain discrete set of numbers.

References

Louis Boutet de Monvel and François Trèves, On a class of pseudodifferential operators with double characteristics, Invent. Math. 24 (1974), 1–34. MR 353064, DOI 10.1007/BF01418785
Louis Boutet de Monvel and François Trèves, On a class of systems of pseudodifferential equations with double characteristics, Comm. Pure Appl. Math. 27 (1974), 59–89. MR 350232, DOI 10.1002/cpa.3160270105
M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math. 26 (1973), 455–475. MR 338532, DOI 10.1002/cpa.3160260403
Antonio Gilioli and François Trèves, An example in the solvability theory of linear PDE’s, Amer. J. Math. 96 (1974), 367–385. MR 355285, DOI 10.2307/2373639
Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York, Inc., New York, 1969. Third revised printing. MR 0248435
A. Menikoff, Uniqueness of the Cauchy problem for a class of partial differential equations with double characteristics, Indiana Univ. Math. J. 25 (1976), no. 1, 1–21. MR 399626, DOI 10.1512/iumj.1976.25.25001
François Trèves, Concatenations of second-order evolution equations applied to local solvability and hypoellipticity, Comm. Pure Appl. Math. 26 (1973), 201–250. MR 340804, DOI 10.1002/cpa.3160260206

Linear partial differential equations with constant coefficients

A link between local solvability of pseudodifferential operators and uniqueness in the Cauchy problem

94

François Trèves, Discrete phenomena in uniqueness in the Cauchy problem, Proc. Amer. Math. Soc. 46 (1974), 229–233. MR 352679, DOI 10.1090/S0002-9939-1974-0352679-5
M. H. Protter, The Cauchy problem for a hyperbolic second order equation with data on the parabolic line, Canad. J. Math. 6 (1954), 542–553. MR 64269, DOI 10.4153/cjm-1954-059-x

Elliptic pseudodifferential operators on a closed manifold which degenerate on a submanifold

10