Abstract
Existence of a weak solution for some boundary value problems of mixed and hyperbolic types is proved by constructing multipliers which satisfy the multiplier inequalities. The nature of a singularity is then examined by utilizing norm estimates obtained by multipliers. A weak solution of a model hyperbolic problem is shown to have, in an integral sense, a logarithmic singularity at one parabolic point. A model mixed-type problem exhibits a fundamental-type singularity, when solved numerically. In general, it is very difficult to construct multipliers for an arbitrary mixed-type equation and an arbitrary domain. The multipliers constructed in this study satisfy the multiplier inequalities only for a local problem, i.e., the hyperbolic boundary should satisfy a particular condition near the parabolic points and the elliptic boundary should be close to the parabolic line. In order to apply these multipliers to a practical problem, an admissible domain was modified in a δ-neighborhood of the parabolic points. This is called a geometric regularization. By using compact operator theory, the solution is extended further into the general elliptic domain. Note that the local geometry converges to the original geometry as δ approaches to zero. Hence, it is conjectured that the nature of the singularity is retained, and in the limit is the same order as the fundamental solution...
Kim, Young Sook (1990). Analytical and numerical methods for boundary value problems of mixed type. Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -1117091.