Question

A function f (x, y) with continuous second partial derivatives is called harmonic, if it

 ∂²f/ ∂x² + ∂²f/∂y²=0

(a) Suppose that z = f (x, y) is harmonic and that ∂²f ∂ x² (x₀ , y₀ ) ≠ 0. Prove that f (x, y) cannot have a local maximum or minimum at (x₀ , y₀ ). 

(b) Conclude that if f (x, y) is harmonic on the region x² + y² < 1 and is zero on x² + y² = 1 and additionally ∂² f ∂ x² (x, y) ≠0 for all points (x, y), then f (x, y) is zero everywhere on the unit disc.

Answer 1

A function f (x, y) with continuous second partial derivatives is called harmonic, if it satisfies 

∂² f /∂ x² + ∂²f /∂y² = 0 . 

(a) We consider the second derivative test. We have

The identity C = −A follows, because we have assumed that f (x, y) is harmonic. Therefore 

AC − B² = −A² − B² < 0 .

Note that because we assumed that A≠ 0, the quantity AC − B² is strictly negative and therefore any critical point has to be a saddle point. In particular f (x, y) cannot have a local maximum at (x₀ , y₀ ).

(b) If f (x, y) is a function on the disc, then it attains the minimum and maximum at some point. We have seen above, that this point cannot lie in the interior of the disc, because all critical points in the interiour are saddle points. Therefore both the minimum and maximum have to lie on the boundary of the disc, but the function is identically zero on the boundary. Hence the function is identically zero everywhere.