Vol. 63, No. 1, 1976

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Gevrey classes and hypoelliptic boundary value problems

Ralph Artino

Vol. 63 (1976), No. 1, 1–21

Let P(D,Dt) be a hypoelliptic differential operator with constant coefficients of type μ with index of hypoellipticity equal to d 1. Let Ω be an open subset of the half space R+n+1 with plane piece of boundary ω contained in R0n. Let Q1(D,Dt),,Qμ(D,Dt) be μ partial differential operators with constant coefficients and consider the boundary value problem:

  P (D, Dt)u = f  in Ω
Qν(D,Dt)u|ω = gν 1 ≦ ν ≦ μ.

In this paper necessary and sufficient conditions are given on Q1,,Qμ in order that all solutions u of (1) shall belong to the Gevrey class of index d in Ω ω whenever the initial data belong to such classes of functions. In particular, we give not only algebraic conditions but also show how to construct a parametrix for such problems.

Mathematical Subject Classification 2000
Primary: 35H05, 35H05
Secondary: 35E99
Received: 30 March 1972
Revised: 27 October 1975
Published: 1 March 1976
Ralph Artino