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.