A Harmonic Function is Zero on an open portion of the boundary, help

In summary, the conversation discusses a problem involving a harmonic function in a region with specific boundary conditions. The person is struggling to solve the problem and has tried for hours without success. Someone suggests using the Cauchy Kovelevskaya theorem, but it turns out to be flawed. Another person mentions Gronwall's inequality as a possible solution, but later reveals it was a joke. The conversation ends with a question about whether anyone has found a solution to the problem.
  • #1
lackrange
20
0
A harmonic function in a region is zero on an open portion of the boundary, and its normal derivative is also zero on the same part, and it is continuously differentiable on the boundary. I have to show that the function is zero everywhere, but I have no idea how. I have tried this for hours and hours, and haven't come up with anything useful. The best answer I've had so far involves the Cauchy Kovelevskaya theorem, but that was shown to be flawed. Can anyone help? This is very difficult..

I'm pretty sure this implies the gradient is zero on the boundary, is there anything I can possible do with that?
 
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  • #2
Hi lackrange,

This is actually a pretty easy corollary to Gronwall's inequality. Just play around with parameters β as shown on the wikipedia page
 
  • #3
Actually I'm just trolling, I think Gronwall's inequality is completely irrelevant to this problem but it has a really cool name. This post came up after a google search on the day before my takehome midterm was due, and I assumed you were writing the same takehome as me since that question was on the test. I didn't think it was fair that you were asking people on the internet to solve your takehome midterm for you so I posted that reply as as joke.

... This didn't turn out as funny as I expected since I thought you'd reply asking me what the hell Gronwall's inequality has anything to do with this and then I'd tell you that it was all a joke. So I'm sorry if this caused any confusion to anyone, mods feel free to delete these two posts.
 
  • #4
Do you guys know the solution finally?
 

1. What is a harmonic function?

A harmonic function is a type of function in mathematics that satisfies the Laplace equation, which is a second-order partial differential equation. It is commonly used in physics and engineering to model physical phenomena such as heat flow, fluid dynamics, and electromagnetism.

2. What is the boundary in the context of a harmonic function?

The boundary in the context of a harmonic function refers to the edge or boundary of a given domain or region in which the function is defined. It can be a physical boundary, such as the walls of a room, or an abstract boundary, such as the edge of a circle.

3. Why is a harmonic function zero on an open portion of the boundary?

This is because of a property known as the mean value property, which states that the average value of a harmonic function over any closed surface is equal to the value of the function at any point on that surface. If the harmonic function is zero on an open portion of the boundary, it means that the average value over that portion is also zero.

4. What does it mean for a harmonic function to be zero on an open portion of the boundary?

When a harmonic function is zero on an open portion of the boundary, it means that the function takes on a value of zero at every point on that portion of the boundary. This can have important implications in solving differential equations and modeling physical systems.

5. How is the concept of a harmonic function used in real-world applications?

Harmonic functions are used in a variety of real-world applications, including in physics, engineering, and computer graphics. They are used to model and analyze physical phenomena such as heat transfer, fluid flow, and electrical fields. In computer graphics, harmonic functions are used to generate smooth surfaces and textures, creating realistic and visually appealing images.

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