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LE MATEMATICHE Vol. LX (2005) – Fasc. II, pp. 267–279
WEAK WELL-POSEDNESS OF THE DIRICHLET PROBLEM FOR EQUATIONS OF MIXED ELLIPTIC-HYPERBOLIC TYPE KEVIN R. PAYNE Equations of mixed elliptic-hyperbolic type with a homogeneous Dirichlet condition imposed on the entire boundary will be discussed. Such closed problems are typically overdetermined in spaces of classical solutions in contrast to the well-posedness for classical solutions that can result from opening the boundary by prescribing the boundary condition only on a proper subset of the boundary. Closed problems arise, for example, in models of transonic fluid flow about a given profile, but very little is known on the wellposedness in spaces of weak solutions. We present recent progress, obtained in collaboration with D. Lupo and C.S. Morawetz, on the well-posedness in weighted Sobolev spaces as well as the beginnings of a regularity theory.
1. Introduction. The purpose of this note is to examine the question of well-posedness for the Dirichlet problem for a second order linear partial differential equation of mixed elliptic-hyperbolic type. That is, given f ∈ H0 , we ask if it is possible to show the existence of a unique u ∈ H1 which solves in some reasonable sense the problem (1.1)
Lu = K (y)u xx + u yy = f in
Supported by MIUR, Project “Metodi Variazionali e Topologici nello Studio di Fenomeni Non Lineari” and MIUR, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
268 (1.2)
KEVIN R. PAYNE
u = 0 on ∂
where H0 , H1 are functions spaces to be determined, K ∈ C 1 (R2 ) satisfies (1.3)
K (0) = 0 and y K (y) > 0 for y = 0,
is a bounded open and connected subset of R2 with piecewise C 1 boundary. We will assume throughout that (1.4)
± := ∩ R2± = ∅,
so that (1.1) is of mixed elliptic-hyperbolic type. We will ca