In these lectures we study the boundary value problems associated with elliptic
equation by using essentially L^2 estimates (or abstract analogues of such estimates.
We consider only linear problem, and we do not study the Schauder estimates.
We give first a general theory of “weak” boundary value problems for elliptic operators. (We do not study the non-continuous sesquilinear forms; of. Visik, Lions, Visik-Ladyzeuskaya).
We study then the problems of regularity-firstly regularity in the interior, and secondly the more difficult question of regularity at the boundary. We use the Nirenberg method for Dirichlet and Neumann problems and for more general cases we use an additional idea of Aronsazajn-Smith.
These results are applied to the study of new boundary problems: the problems
of Visik-Soboleff. These problems are related and generalize the problems
of the Italian School (cf. Magenes).
We conclude with the study of the Green’s kernels, some indications on
unsolved problems and short study of systems. Due to lack of time we have
not studied the work of Schechter nor the work ofMorrey-Niren-berg
which rots essentially on L^2 estimates. |