An iff solvability condition for the oblique derivative problem. (English) Zbl 0737.35171

Lerner, N.

Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1990-1991, No.XVIII, 7 p. (1991).

The purpose of this note is to prove the following Theorem: For the oblique derivative problem, condition \((\psi)\) is equivalent to solvability. — As a matter of fact, the proof is a modification of the argument we used to handle the two dimensional case of the Nirenberg- Treves conjecture for pseudodifferential operators in [Ann. of Math. 128, 243-258 (1988; Zbl 0682.35112)]. The basic remark is that the oblique derivative problem is equivalent to a pseudodifferential equation of a very particular type \(\partial_ t+\alpha(t,x)\Omega(t,x,D_ x)\) where \(\Omega\) is a non-negative pseudodifferential operator and \(\alpha\) a smooth function.

MSC:

35S15	Boundary value problems for PDEs with pseudodifferential operators
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

local solvability; condition \((\psi)\); Nirenberg-Treves conjecture; pseudodifferential operators; oblique derivative