Question

A function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is said to be harmonic if all its second partial derivatives are continuous and \(D_{11} f+D_{22} f=0\). (i) What functions of the form \(f(s, t)=a s^{2}+2 b s t+c t^{2}\) are harmonic? (ii) What can be deduced about the points \(z\) at which a harmonic function \(f\) satisfies \(\left(D_{1} f\right)(z)=\left(D_{2} f\right)(z)=0\) ? (iii) If \(U=\left\\{(s, t): s^{2}+t^{2} \leq 1\right\\}\), what can be said about the points where \(\sup f(U)\) and inf \(f(U)\) are attained?

Short answer

Answer: At points z where \((D_1 f)(z) = (D_2 f)(z) = 0\), they are either saddle points or local extrema points of the harmonic function. The suprema and infima of the given harmonic function occur at the boundary of the domain \(U\).

Step-by-step solution

(i): Finding harmonic functions

In this part, we need to find the second partial derivatives of the given function and check if they satisfy the harmonic condition. The function is given by: \(f(s, t) = as^2 + 2bst + ct^2\) First, find the first partial derivatives: \(D_1 f = \frac{\partial f}{\partial s} = 2as + 2bt\) \(D_2 f = \frac{\partial f}{\partial t} = 2bs + 2ct\) Now, find the second partial derivatives: \(D_{11}f = \frac{\partial^2 f}{\partial s^2} = 2a\) \(D_{22}f = \frac{\partial^2 f}{\partial t^2} = 2c\) Since both are constants, they are continuous. Now, check if the function is harmonic: \(D_{11}f + D_{22}f = 2a + 2c = 0\); so \(a = -c\) Hence, harmonic functions of the given form are those satisfying \(a = -c\).

(ii): Deduction about points z

Given a harmonic function f, we want to deduce information about points z where \((D_1 f)(z) = (D_2 f)(z) = 0\). At these points, its gradient vector is equal to the zero vector. Specifically, we have: \((D_1 f)(z) = 0 \Rightarrow 2az_1 + 2bz_2 = 0\) \((D_2 f)(z) = 0 \Rightarrow 2bz_1 + 2cz_2 = 0\) Since the function is harmonic, \(a = -c\), and we can write the previous equations as: \((-c) z_1 + bz_2 = 0\) \(b z_1 - c z_2 = 0\) In addition, as f is harmonic, the Laplacian at points z equals zero. Hence, points z are saddle points of the harmonic function f or local extrema points.

(iii): Points where sup(f(U)) and inf(f(U)) are attained

As \(f\) is harmonic, it satisfies the mean-value property; that is, the value of the function at any point \((s, t)\in U\) equals the average value of the function on any circle centered at \((s, t)\) and lie entirely in \(U\). This implies that at the point where sup(f(U)) is attained, the average value of the function on any circle centered at this point and lying entirely in \(U\) is greater than or equal to the values of the function at other points in \(U\). Similarly, at the point where inf(f(U)) is attained, the average value of the function on any circle centered at this point and lying entirely in \(U\) must be less than or equal to the values of the function at other points in \(U\). Since both the sup and inf cannot be raised merely by changing the values at a single point, the suprema and infima must be attained at the boundary of the domain \(U\). So, we can conclude that the points where sup(f(U)) and inf(f(U)) are attained belong to the boundary of \(U\).

Key concepts

Partial Derivatives

In the context of harmonic functions, partial derivatives play a pivotal role. These derivatives are essentially the rates at which a function changes with respect to one variable while keeping all other variables constant. For example, in a two-dimensional function like our example above, the first partial derivatives with respect to the variables s and t represent how the surface slopes in the respective variable's direction. Once we have the first partial derivatives, finding the second partial derivatives is just a matter of differentiation.

For harmonic functions, we're particularly interested in the second-order partial derivatives because their sum, known as the Laplacian, determines if a function is harmonic. If these derivatives satisfy the condition that their sum equals zero, the function exhibits the harmonious balance of no net curvature, like the surface of a drum vibrating in equilibrium.

Laplacian

The Laplacian is a critical operator in the study of harmonic functions. It essentially gives us the sum of all second partial derivatives, indicating the function's concavity in all directions. For a function f(s, t), the Laplacian is denoted by \(\Delta f = D_{11}f + D_{22}f\). When this value is zero throughout the domain of f, the function is deemed harmonic.

The use of the Laplacian in the step-by-step solution sheds light on the inherent property of harmonic functions: they are local mean approximations of their surroundings. This is why checking if a function is harmonic involves evaluating its Laplacian and ensuring that it indeed equals zero.

Mean-Value Property

The mean-value property is a distinctive characteristic of harmonic functions. It states that the value of a harmonic function at a certain point is the average of the function's values over a sphere or circle centered at that point.

In layman's terms, if you could magically place a perfectly symmetrical soap bubble on any point on the surface described by a harmonic function, the height of the bubble's center represents the average height of all points on the bubble's surface. This property is why harmonic functions do not exhibit extreme values in open domains, as noted in part (iii) of our exercise: the highest and lowest values must occur at the boundaries, because inside, the values are always smoothed out by their neighbors.

Extrema Points

Identifying extrema points, or points at which a function reaches a local minimum or maximum, is a fundamental aspect of calculus. In our context, these are the points of interest in a harmonic function where its gradient equals zero. However, due to the nature of harmonic functions, these points do not manifest as typical peak or valley points. Instead, they could be saddle points - points where the function curves up in one direction and down in another.

This behavior is derived from the fact that a harmonic function is balanced by its Laplacian being zero, essentially not allowing for any net concavity or convexity at any point in its domain, as inferred in part (ii) of our exercise. The extrema, in the case of harmonic functions contained within a bounded domain, are particularly interesting because, by the mean-value property, they are bound to occur on the periphery of the domain.