University Mathematics Handbook (2015)

XIV. Partial Differential Equations (PDE)

Chapter 7. Laplace Equation

7.1  Introduction

Let be a domain with boundary , a normal to , and a Laplace operator

a.  Finding solution in of Laplace equation .

with the boundary condition:

1.   Dirichlet problem

2.   Neumann problem

3.   Robin problem

b.  The Dirichlet problem has a unique solution; the Neumann problem has a unique up to constant solution; the Robin problem has a unique solution when and is not identically zero.

c.  Maximum Principle: If is a harmonic function (see XII.4.5), bounded on , and continuous on then attains its maximum on .

7.2  Dirichlet Problem in a Rectangle

The solution is a superposition of 4 functions, , each function being a solution of a Dirichlet problem where boundary condition, except , is identically zero.

Example: If , the problem is solved by separation of variables .

The generalized solution is when are Fourier coefficients of , and therefore

7.3  Dirichlet Problem in a Disc

In polar coordinates , , , we get

This problem is solved by separation of variables

when , which leads to , .

The final solution is

 (*)

when and are Fourier coefficients of on

,

(*) can be written in the form of Poisson's integral formula:

.

7.4  Neumann Problem in a Disc

, when

The solution is

when is a constant.