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Weak well posedness of the Dirichlet problem for equations of mixed ellptic-hyperbolic type

Author(s): Kevin R. Payne

Journal: Le Matematiche
ISSN 0373-3505

Volume: 60;
Issue: 2;
Start page: 315;
Date: 2005;
Original page

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 well-posedness 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.
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