can you do this without Laplace and only use definition of harmonic on G

Yes, I can help you solve a problem using the definition of harmonic functions without involving the Laplace operator explicitly. However, to provide specific assistance, I'll need more details about the problem you're referring to. 

In the context of harmonic functions, the general idea is that a function \( u(x) \) is harmonic in a domain \( G \) if it satisfies the mean value property, which can be expressed as:

\[
u(x) = \frac{1}{|\partial B(x, r)|} \int_{\partial B(x, r)} u(y) \, dS(y)
\]
for all \( x \in G \) and for every ball \( B(x, r) \subset G \), where \( \partial B(x, r) \) is the boundary of the ball and \( dS(y) \) is the surface measure.

If you can share the specific harmonic problem or function you're dealing with, I can guide you through the process step by step using the definition and properties of harmonic functions.

Would you like to provide the specific function or problem? Feel free to upload a photo or provide more details.

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### 5 Related Questions:
1. How can we verify whether a function is harmonic using the mean value property?
2. What are the boundary conditions that typically accompany harmonic functions?
3. How does the maximum principle apply to harmonic functions on a domain \( G \)?
4. How does the definition of harmonic functions differ from the Laplace equation?
5. Can we prove that a function is harmonic by using Green’s theorem or other integral methods?

### Tip:
Harmonic functions are smooth (infinitely differentiable) within their domain, which can often simplify checking their properties without using advanced differential operators.

Discover how to solve harmonic functions on a domain G using the mean value property, without relying on the Laplace operator. This method involves fundamental concepts like the mean value property and the maximum principle for harmonic functions, commonly taught at the undergraduate level in advanced calculus.

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