# How does Laplace's equation ensure that the condition of saddle (i.e. the Hessian have eigenvalues of mixed sign) is satisfied?

For a function $$f(x,y,z)$$ that satisfies the Laplace's equation, $$\nabla^2f=0,$$ every point must be a saddle point (because it allows no local maxima or minima). Therefore, the condition $$\nabla^2f=0$$ must be sufficient to ensure that the Hessian matrix $$\Delta(x,y,z)= \begin{bmatrix} f_{xx} & f_{xy} & f_{xz}\\ f_{yx} & f_{yy} & f_{yz}\\ f_{zx} & f_{zy} & f_{zz}\\ \end{bmatrix}$$ has both positive and negative eigenvalues.

First I want to know whether what I have said above, is correct. If yes, how can we prove that the eigenvalues of $$\Delta(x,y,z)$$ indeed have mixed signs (both positive and negative) assuming $$\nabla^2f=0$$.

• By definition, saddle points are critical points of the function? Commented Apr 16, 2022 at 22:16

Any solution of the Laplace equation $$\nabla^2 f = 0$$ is called a harmonic function, which satisfies the strong maximum principle: Unless $$f$$ is a constant function, the maximum (or minimum) of $$f$$ cannot also be achieved anywhere in the interior of the domain. Thus, any stationary point or a critical point of a harmonic function $$f$$ is a saddle point.