The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

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How can I mathematically proof an incoherent superposition of waves?

Let $\psi = A(t)\cos(\theta_1(t))$ and $\phi = B(t)\cos(\theta_2(t))$ two independent waves which phases and amplitudes depend on the time. Then it follows that the intensity of the superposition of ...
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Dirichlet with $C^1$ boundary data

Do you know an explicit example for an harmonic function $u$ in a regular domain $\Omega$ which satisfies the following two conditions? $u$ belongs to $C^0(\overline{\Omega})$ but not to ...
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An identity for the integral $ \int_{\partial B(0,1)}u(x_0+aw)u(x_0+cw)$ with a harmonic function $u$

This is Question 2.18 from Gilbarg and Trudinger, chapter 2. We are given that $\Omega$ is open bounded smooth boundary. Now fix $x_0\in \Omega$ and a constant $c>0$ such that ...
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Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
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15 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
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Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
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1answer
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Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
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37 views

Analytic function on the whole plane, positive imaginary part, what can it be?

Part (a): The function f is analytic in the whole plane with positive imaginary part. What can it be? Part (b): What if all you know is that the imaginary part of f tends to 0 at infinity? what we ...
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Harmonic function with boundary value 0 except one point

Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$. (a) Find an example ...
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1answer
65 views

You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
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27 views

Prove that harmonic numbers satisfy the equality. [duplicate]

The $k$th harmonic number is defined to be $H_k = 1 + 1/2+ 1/3 + · · · + 1/k .$ Prove that harmonic numbers satisfy the equality $H_1 + H_2 + · · · + H_n = (n + 1)H_n − n$ for all $n \in\Bbb N.$
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Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
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2answers
58 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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21 views

Harmonic function with Neumann BC

I have a problem where I denote by $D = B(0, 2)$ the disk in the plane with radius 2 centered at the origin. I have to find a harmonic function $u(r, \theta)$ in $D$ which satisfies the additional ...
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67 views

Isolated singularity of harmonic function

I am working with a book by Axler, Bourdon and Ramey and find the following problem: Suppose $u$ is a harmonic function on $B \setminus \{0 \}$ such that $$ |x|^{n-2} u(x) \to 0, \qquad x \to 0 $$ ...
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1answer
19 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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1answer
14 views

Harmonicity of the expectation of a stopped Brownian Motion

Let $\mathbb{E}_x$ be the expectation associated with a probability measure such that $B_{t\geq0}$ is a Brownian motion started in x. I want to show that for $D\subset\mathbb{R}^2$ bounded, $y\in D, ...
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Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
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Poisson Integral, when $U$ is discontinuous

So I am working on the following problem. Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one ...
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1answer
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Dirichlet Problem

I have to solve the following Dirichlet Problem $$\Delta u=0\quad\text{in}\,\,\, D,$$ $$u(\mathrm{e}^{it})=\frac{1}{2}(\mathrm{e}^{it}+\mathrm{e}^{-it}),$$ for $$u \in C^2(D)\cap C(\overline{D}).$$ ...
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My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
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35 views

Prove that $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic

Suppose that $u,v$ are harmonic functions on doman $D$, and they are harmonic conjugate. Prove that function $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic on $D$. What I've done was to take the ...
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Area growth of harmonic functions

Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?
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Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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Why is the Laplace/Helmholtz equation only separable in a finite number of coordinate systems?

On MathWorld one finds that the Helmholtz equation $$(\nabla^2+k^2)\psi=0$$ is only separable in 11 coordinate systems. Similar statements can be found about the Laplace equation and maybe other ...
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Can the measure of zeroes of a harmonic function be positive?

Let $u$ be a non-constant harmonic function of two variables defined, say, in the unit disk (or on the half plane for example). It is known that $u$ can vanish on some lines, as it discussed in here. ...
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1answer
24 views

Harmonic conjugate extend to boundary [duplicate]

Suppose u is a harmonic function in disc $|z|<1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary? I know v can ...
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1answer
55 views

Can a real harmonic function on the unit disk satisfy $f(0)=1$ while the area of $\{z:f(z)>0\}$ is zero?

Does there exist a harmonic function defined in the unit disk such that (1) $f(0)=1$ (2) area of $\{z\in\mathbb{D}:\,f(z)>0\}$ is equal to zero? I tried to use certain representations of ...
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1answer
30 views

Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
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38 views

Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...
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Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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Uniqueness in boundary value problem for the biharmonic functions

My attempt: I tried to use the Green's representation formula twice. The Green's reprensentation formula:$u(y)=\int_{\partial \Omega}(u(x)\frac{\partial G(x-y)}{\partial v}-G(x-y)\frac {\partial ...
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1answer
39 views

Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
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1answer
42 views

The solution of $\Delta u=u^3$ with zero boundary values is identically zero

My question: My attempt: I tried to use the Representation using Green's formula: Since $u=0$ on the boundary and $f(x)=x^3$, then the formula becomes: $$u(x)=\int_\Omega y^3G(x,y)dy \quad ...
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Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...
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1answer
41 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
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1answer
54 views

Proof of reflection principle for harmonic functions

** My attempt: Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap ...
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A harmonic function bounded from below is constant

I am learning PDE on myself as a beginner. It takes me like several hours to finally think out this proof. However, I feel something not right about my proof, especially choosing "$R$" part, it ...
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1answer
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Prove that $(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$

Knowing: $f(z)$ is analytical Prove: $$(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$$ I have proved firstly that $\ln|f(z)|$ is harmonic function Let ...
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1answer
77 views

Maximum principle for harmonic functions on unbounded domain

I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle. My attempt: Using the $\lim u(x)=0$, then exists $\epsilon$ and $N$, ...
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1answer
39 views

Solution of Dirichlet problem in a ball?

If $u\in C^2(B_{R}(0))\cap C^0(B_R(0)))$ is harmonic, then $\large u(x)=\frac{R^2-|x|^2}{nW_nR}\int_{\partial B_R(0)}\frac{u(y)}{|x-y|^n}\,ds(y)=\int_{\partial B_R(0)}k(x,y)u(y)\,ds(y)$ by the ...
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1answer
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How to do the “direct calculation”?

How I can get the equation (2.25)? What is the calculation? If $|x|$ is the length of the vector, I can't see how to do partial differential with respect to a length? Can someone help me? It is ...
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3answers
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evaluate the sum of an alternating harmonic series with a fixed limit

Hi I stumbled across an alternating harmonic sum with a fixed limit on an practice exam and I've no idea how to calculate this sum. \begin{equation} \sum_{1}^{100}\frac{(-1)^{k-1}}{k}. \tag{1} ...
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39 views

Show $\phi \circ f$ is subharmonic.

Prove that if $f:G\rightarrow \Omega$ is a one-to-one holomorphic function and $\phi:\Omega \rightarrow \mathbb{R}$ is a smooth (twice continuously differentiable) subharmonic function, then $\phi ...
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1answer
29 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
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1answer
26 views

Help with proof about sub harmonic function

I know how to prove it using strong maximum principle, but I need to show it using conditions for a relative maximum. Does this mean using second derivative test? I think if $p\in \Omega$ and ...
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1answer
38 views

A function satisfying the mean value property is harmonic

Here is the problem. I know that if $u$ is harmonic the equation holds, but I don't know how to prove it from the other direction.
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33 views

Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.

Is the form correct for the conjugate harmonic? Attempt: First, we are given \begin{align*} \log \left| z \right| &= u(x,y) + iv(x,y) = \log \sqrt{x^2 + y^2} + i \cdot 0 \\ u(x,y) &= \log ...
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Properties of harmonic functions

If $u:D \mapsto \mathbb{R}$ and $v:D \mapsto \mathbb{R}$ are harmonic functions, then also function $uv:D \mapsto \mathbb{R}$ is harmonic. Is the statement correct?