Approximation by Analytic and Harmonic Functions, Incompressible Vector Fields, and Temperature Distributions

The best in the L2 sense.approximations of a given function by solutions of certain PDEs are found, including approximations by analytic and by harmonic functions, by incompressible vector fields, and by solutions of the heat equation. In each case the problem is reduced to a boundary value problem BVP.for a second or fourth order PDE. The corresponding BVPs are well posed and can be solved numerically. For some domains, analytic expressions for the solutions are found.