为何$u=\log(\sqrt{x^2+y^2})$在$x^2 + y^2 <1$内不调和？奇点疑问

Hey there! Let's tackle your two questions about harmonic functions with clear, straightforward explanations.

First, let's recall the core definition of a harmonic function: a function is harmonic on a region if it’s twice continuously differentiable at every single point in that region, and satisfies the Laplace equation $\Delta u = u_{xx} + u_{yy} = 0$ everywhere within it.

Now look at $u=\log(\sqrt{x^2+y2})$ — this simplifies to $\frac{1}{2}\log(x^2+y2)$. The problem here is the origin $(0,0)$, which is fully inside the region $x^2+y2 <1$. At the origin:

• The function isn’t even defined (since $\log(0)$ is undefined, and the limit as $(x,y)\to(0,0)$ blows up to $-\infty$).
• You can’t talk about continuity, let alone second-order derivatives, at a point where the function doesn’t exist in the first place.

Since harmonicity requires the entire region to meet the definition, and this region includes a point where the function fails on the most basic level (being defined), it can’t be harmonic on $x^2+y2 <1$.

Let's split this into two easy-to-digest parts:

Why it's harmonic when $x,y > 0$, but not at $(0,0)$ #

When $x,y > 0$, we’re working with a region that excludes the origin. In every point of this region:

• The function is well-defined (since $x^2+y2 >0$), continuous, and twice continuously differentiable.
• If you compute the Laplacian step by step:
  • First partial derivatives: $u_x = \frac{x}{x^2+y2}$, $u_y = \frac{y}{x^2+y2}$
  • Second partial derivatives: $u_{xx} = \frac{y^2 - x^2}{(x2+y²⁾2}$, $u_{yy} = \frac{x^2 - y^2}{(x2+y²⁾2}$
  • Adding them gives $\Delta u = \frac{(y^2 - x^2) + (x^2 - y^2)}{(x2+y²⁾2} = 0$, which checks out for the Laplace equation.

At $(0,0)$, though, the function is undefined, and there’s no way to assign a value to make it continuous there (the limit is infinite). Since harmonicity requires the function to be twice differentiable at the point, $(0,0)$ can never be a point where this function is harmonic.

Conditions for harmonic functions without singularities #

A harmonic function has no singularities on its domain if:

1. Its domain is an open, connected region where the function is defined everywhere.
2. The function is twice continuously differentiable at every point in the domain.
3. It satisfies the Laplace equation $\Delta u =0$ at every point in the domain.

Singularities pop up when any of these conditions fail. Here are the two main types:

• Removable singularities: If a function is harmonic everywhere in a region except one point, and the limit of the function at that point exists (and is finite), you can extend the function to that point to make it harmonic over the larger region.
• Non-removable singularities: Like the origin for $\frac{1}{2}\log(x^2+y2)$ — the limit at the origin is infinite, so no matter how you define the function there, it can’t be continuous (let alone twice differentiable). These are "essential" singularities that can’t be fixed to make the function harmonic at that point.

In short: A harmonic function avoids singularities if its domain doesn’t include points where the function is undefined, discontinuous, or fails to have continuous second derivatives that satisfy the Laplace equation.

内容的提问来源于stack exchange，提问作者villanif