Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 376

Extracting its real part leads to the solution of PDS (7.83), by noting that u | r = R =

f ( θ ) ,

2π

R 2 r 2 1

2

−

( θ ) θ ,

u = f d

R 2 r 2 ( θ − θ )

π + − 2 Rr cos 0

which is the same as Eq. (7.17).

7.5 Well-Posedness of Boundary-Value Problems

Since boundary-value problems of nonhomogeneous potential equations can be
transformed to those of homogeneous equations, we only discuss the well-posedness
of boundary-value problems of Laplace equations in this section.

Theorem 1. If a Dirichlet problem has a solution, the solution must be unique and
stable.

Proof. Uniqueness: Let u 1 and u 2 be two solutions of a Dirichlet problem of the
Laplace equation. Then u = u 1 − u 2 must satisfy

Δ

u = 0 , Ω ,

(7.84)

u | ∂Ω = 0 .

By the extremum principle, we obtain

u ≡ 0 , Ω ,

so that u 1 ≡ u 2 . By the arbitrariness of u 1 and u 2 , we establish the uniqueness of the

solution.

Stability: Let u 1 and u 2 be the solutions of
⎧
⎨
⎩

Δ u = 0 , Ω ,

= ,

Ω ,

Δ u 0

and

u | ∂Ω = f 1

u | ∂Ω = f 2 ,

respectively. u = u 1 − u 2 thus must satisfy

⎧
⎨

Δ u = 0 , Ω ,

⎩ u | ∂Ω = f 1 − f 2 .

By the extremum principle,

sup
¯
Ω

| u | = sup

∂Ω |

f 1 − f 2 | = sup

¯
Ω

| u 1 − u 2 | .

Thus, if sup
∂Ω

| f 1 − f 2 | < ε ,sup

Ω

| u 1 − u 2 | < ε , so the solution is stable with respect to

the boundary values.