Krein [1] mentioned that for each PD equation we have two extreme operators, one is the minimal in which solution and its derivatives on the boundary are zero, the other one is the maximal operator in which there is no prescribed boundary conditions. They claim it is not possible to have a related boundary value problem for an arbitrarily chosen operator in between. They have only considered lo...

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order e...

The main part of this chapter is devoted to the development and presentation of a coupled multigrid method for the solution of saddle point systems (2.51) arising from the discretization of PDE constrained optimization problems. In subsequent chapters, the devised method will be adapted to handle inequality constraints on the control, and it will be employed for the solution of the systems, whi...

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography that appears as an important application in Underwater Acoustics, we analyze a general Schrödinger-type equation posed on two-dimensional variable domains with mixed boundary conditions. The resulting initialand boundary-value problem is transformed into an equivalent one posed on a ...

Sharp estimates for the solutions to elliptic PDEs in L∞ in terms of the corresponding norm of the boundary data follow directly from the maximum principle. It holds on arbitrary domains for all (real) second order divergence form elliptic operators −divA∇. The wellposedness of boundary problems in L, p < ∞, is a far more intricate and challenging question, even in a half-space. In particular, ...