Existence and uniqueness theorems for the solution of an inverse problem for the transport equation (Q1092240)

The author studies some direct and inverse problems for the transport equation \[ u_ x \sin \phi +u_ y \cos \phi +u_{\phi}f=\lambda \] (u\(=u(x,y,\phi)\), \((x,y)\in D\subset {\mathbb{R}}^ 2\), \(\phi\in (0,2\pi))\), for example that of obtaining u and \(\lambda\) if f and the restriction of u to \(\partial D\times (0,2\pi)\) are given. Uniqueness theorems are proved by means of a priori estimates, existence theorems by a combination of regularization and Galerkin approximations. Here the two- and three-dimensional cases, however, require different techniques (e.g. different a priori estimates and choices of the function spaces involved).