Initial-boundary value problems for linear PDES: The analyticity approach

It is well-known that one of the main difficulties associated with any method of solution of initial-boundary value problems for linear PDEs is due to the presence of boundary values which cannot be arbitrarily assigned. To deal efficiently with this difficulty, we have recently proposed two alternative (but interrelated) methods in Fourier space: the Analyticity approach and the Elimination by Restriction approach. In this work we present the Analyticity approach and we illustrate its power by studying the well-posedness of initial-boundary value problems for second and third order evolutionary PDEs, and by constructing their solution. We also show the connection between the Analyticity approach and the Elimination by Restriction approach in the particular case of the Dirichlet and Neumann problems for the Schrodinger equation in the n-dimensional quadrant.