| Abstract: |
| We will discuss the Hadamard well-posedness (existence, uniqueness and continuous dependence on the data) of the nonlinear Schr\{o}dinger equation with power nonlinearity formulated on the spatial quarter-plane with Sobolev initial data and nonhomogeneous Dirichlet boundary data in appropriate Bourgain-type spaces. We will work in a low regularity setting, where Strichartz estimates are necessary in addition to the standard Sobolev estimates. An interesting feature of this problem when compared to related works in the literature is that both of the spatial variables are restricted to the half-line. For this reason, a new approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. An additional challenge is posed by the fact that the spatial domain (quarter-plane) has a corner at the origin, thus requiring one to expand the validity of certain Sobolev extension results to the case of a domain with a non-smooth (Lipschitz) and non-compact boundary. This is joint work with Turker Ozsari. |
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