Solving Laplacian Equation with u ≤ √x

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In summary, the conversation discusses a problem involving Laplacian and a solution provided in a document. The solution involves defining a new variable, v, and using a formula (equation 10) to show that u(x) is bounded by a certain value. However, there seems to be an error in the algebra, specifically in the substitution of v(0) with √R. It is unclear why this substitution was made and how it affects the overall solution.
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PeteSampras
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Homework Statement


Problem 2 http://math.mit.edu/~jspeck/18.152_Spring%202017/Exams/Practice%20Midterm%20Exam.pdf
"Let ##u## such that ##Laplacian( u)=0##
Show if ##u \le \sqrt{x}##, then ##u=0##

Homework Equations


At the solution http://math.mit.edu/~jspeck/18.152_Spring%202017/Exams/Practice%20Midterm%20Exam_Solutions.pdf
define ##v=u + \sqrt{R}##

The Attempt at a Solution



equation 10 says, by Harnack
##\frac{R(R-|x|)}{(R+|x|)^2}v(0) \le v(x) \le \frac{R(R+|x|)}{(R-|x|)^2}v(0) ##

but my question is, why in formula 10 change v(0) by sqrt{R} at the left and right side?
## ( \frac{R(R-|x|)}{(R+|x|)^2}-1) \sqrt{R} \le u(x) \le ( \frac{R(R+|x|)}{(R-|x|)^2}-1) \sqrt{R}##

I understand that change ##v(x) \to u(x) + \sqrt{R}##, but, i don't understand why change v(0) by sqrt{R} at the left and right sides.
 
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  • #2
PeteSampras said:
why change v(0) by sqrt{R} at the left and right sides.
I agree with you, the algebra is flawed. I see no way to fix it.
 

1. What is the Laplacian equation?

The Laplacian equation is a mathematical equation that describes the rate of change of a physical quantity, such as temperature or pressure, in terms of its spatial variation. It is commonly used in fields such as physics, engineering, and mathematics.

2. What is the significance of u ≤ √x in the Laplacian equation?

The condition u ≤ √x in the Laplacian equation signifies that the solution u must be less than or equal to the square root of x. This condition is often used to determine the maximum or minimum value of a function.

3. How is the Laplacian equation solved?

The Laplacian equation can be solved using various methods, including separation of variables, Green's functions, and numerical methods such as finite differences. The specific method used depends on the specific problem and the available resources.

4. What is the role of boundary conditions in solving the Laplacian equation?

Boundary conditions play a crucial role in solving the Laplacian equation as they specify the behavior of the solution at the boundaries of the problem domain. These conditions are essential in obtaining a unique solution to the equation.

5. What are some real-world applications of solving the Laplacian equation with u ≤ √x?

The Laplacian equation with u ≤ √x has numerous applications in fields such as heat transfer, fluid dynamics, and electromagnetism. It is also used in image processing, financial modeling, and computer graphics.

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