Can a Non-Surjective Smooth Harmonic Function on $\Bbb R^2$ Be Non-Constant?

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In summary, a non-surjective smooth harmonic function on $\mathbb{R}^2$ can be non-constant, meaning that there are points in its domain that are not mapped to by the function. A harmonic function is a function that satisfies the Laplace equation and can have a constant value. The difference between a surjective and non-surjective function is that all elements in the output range of a surjective function are mapped to, while not all elements in the output range of a non-surjective function are mapped to. Non-surjective smooth harmonic functions on $\mathbb{R}^2$ have practical applications in various fields, such as physics, engineering, and computer graphics. They can be used to model physical phenomena
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Euge
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Here is this week's POTW:

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Prove that every smooth harmonic function from $\Bbb R^2$ to $\Bbb R$ that is not surjective must be constant.

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Alan receives honorable mention for providing a well thought-out, but incomplete, proof of the result. You can read my solution below.

Let $h : \Bbb R^2 \to \Bbb R$ be a smooth harmonic function that is not surjective. Let $f$ be an entire function whose real part is $h$. Since the image of $h$ misses at least one point $a\in \Bbb R$, the image of $f$ misses at least the line $\operatorname{Re}(z) = a$. By the Little Picard theorem, $f$ must be constant. Therefore, $h$ is constant.
 

FAQ: Can a Non-Surjective Smooth Harmonic Function on $\Bbb R^2$ Be Non-Constant?

Can a non-surjective smooth harmonic function on $\Bbb R^2$ be non-constant?

Yes, it is possible for a non-surjective smooth harmonic function on $\Bbb R^2$ to be non-constant.

What is a smooth harmonic function?

A smooth harmonic function is a function that is both smooth (infinitely differentiable) and harmonic (satisfies Laplace's equation).

What does it mean for a function to be non-surjective?

A non-surjective function is one that does not map to every element in its codomain. In other words, there are elements in the codomain that are not mapped to by the function.

Why is it important for a smooth harmonic function to be non-constant?

A non-constant smooth harmonic function can provide valuable insights into the behavior of functions in mathematics and physics. It can also be used to model various phenomena in the real world.

Are there any real-world applications for non-surjective smooth harmonic functions?

Yes, non-surjective smooth harmonic functions have many real-world applications, such as in fluid dynamics, electromagnetism, and quantum mechanics. They can also be used in image processing and data analysis.

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