0
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0answers
20 views

Integral of harmonic function in a ball

Let $f\in C^2(\Omega)$ an harmonic function in $\Omega$, and: $$ \phi(r) = \frac{1}{2\alpha_2r} \int_{\partial B_r(x)} f(y) d \sigma(y) $$ Prove that $\phi '(r)=0$ by calculating the line integral. ...
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1answer
51 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...
1
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1answer
34 views

The Laplacian and a nice PDE

Given the Laplacian: $$\Delta u= \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} $$ I had to show that by using this $$v(r,\theta ):=u(r\cos \theta ,r\sin \theta ) $$ I can ...
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1answer
34 views

Can we consider the points $(c,y)$ as local minima or maxima for all $y∈ℝ$

Let $f:(a,b)×ℝ→ℝ$ be a non zero and twice differentiable function. Let $c∈(a,b)$. Assume that $Δf=0$ (the Laplacian operator) and $f(c,y)=0$ for all $y∈ℝ$. Assume that $$f(x,y)<0=f(c,y)$$ for ...
2
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1answer
76 views

Boundary integral of a harmonic function around a pole

I have a radial harmonic function $h:\mathbb R^N\backslash\{0\}\to\mathbb R$ which has a pole of order $m$ in 0, and I would like to compute $$ \frac{1}{\sigma_N}\int_{\partial ...
2
votes
1answer
159 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
1
vote
1answer
105 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
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1answer
120 views

Does the Poisson kernel give a unique harmonic function with given boundary data?

In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper ...
4
votes
1answer
234 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
2
votes
1answer
57 views

$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
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1answer
164 views

What is relationship between Wirtinger differential operator and multivariable chain rule?

What is relationship between Wirtinger differential operator(equation 5) and multivarible chain rule(equation 4)? for other Wirtinger related questions look here.
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2answers
278 views

Positive harmonic function on $\mathbb{R}^n$ is a constant?

Is it true that a positive harmonic function on $\mathbb{R}^n$ must be a constant? How might we show this? The mean value property seems not to be the way...for that we would need boundedness.
3
votes
2answers
57 views

Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized

My professor mentioned something like "Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized." I've been trying to understand this statement. If I say that ...
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1answer
173 views

Determining if the product of two particular harmonic functions is a harmonic function

Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$. I would like to show that: $v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$ is also ...
3
votes
1answer
391 views

Is Poisson's/Laplace's equation rotationally invariant?

Is Poisson's equation rotationally invariant? Is Laplace's equation rotationally invariant? If so, how can I see this? in reply to comments, my problem is that: In my notes, when calculating the ...
4
votes
1answer
236 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
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1answer
116 views

Harmonic function product, Knowing that one is Harmonic implies something about the other?

Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea : Let $A$ be an Harmonic ...
2
votes
1answer
281 views

Maximum of strictly subharmonic function

Let $u\in C^2(D)$, $D$ is the closed unit disk in $\mathbf{R}^2$. Assume that $\Delta u>0$. Show that $u$ cannot have a maximum point in $D\setminus\partial D$. This statement is in a calculus ...
3
votes
1answer
251 views

Harmonic function with condition on part of its boundary

Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...