For questions regarding harmonic functions.

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Is the subspace $\{f \in C^{\infty}(\Omega) \;:\; \Delta^nf=0 \;, f=0 \mbox{ on } \Omega \}$ dense? [on hold]

As the title asks, is the subspace $S = \{f \in C^{\infty}(\Omega) \;:\; \Delta^nf=0 \mbox{ for some n, } f=0 \mbox{ on } \partial\Omega \}$ dense in $C^\infty(\Omega)$ (for nice enough $\Omega$)? ...
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28 views

Show that the only nonnegative superharmonic functions in R are the constants

I am having trouble finding g$^∗$(x) when $$g(x) = \begin{cases} xe^{-x} & \text{for x > 0} \\[2ex] 0 & \text{for x $\leq$ 0}. \end{cases}$$ I would like to use the fact that the only ...
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20 views

Proving that only nonnegative superharmonic functions in R$^2$ are constants

How can I prove that the only nonnegative (B$_t$-) superharmonic functions in R$^2$ are the constants? So far, I know that u is a nonnegative superharmonic function and that there exist x, y ∈ R$^2$ ...
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40 views

Intuition regarding $\lim \lVert u_r - u \rVert_{p}=0$

I have some trouble intrepreting the following statment If $u$ is harmonic in $D$ and has bouned means for order p on circles of radius $< 1$ then $\lVert u \rVert_{p}=\lVert u \rVert_{L^{p}}$ ...
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If we know $h_a = H_0^{1}(x + iy),$ for the Hankel function $H_0^{(1)}$, is it possible to determine $h_b = H_0^{1}(x - iy)?$

I am wondering if there is a certain identity for Hankel functions of the first kind of order $0$. If we know $$h_a = H_0^{(1)}(x + iy),$$ where $y > 0$, is it possible to determine $$h_b = H_0^{(...
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Riesz measure in potential theory

I am studying Riesz measures associated to superharmonic funcions, following a book by Doob: Potential theory and its Probabilistic Counterpart. On page 51, the following theorem is introduced: If $u$...
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1answer
34 views

Determining if a function is harmonic in a fast way

Determine which function is harmonic in $\mathbb R^2$: $$\text{a) } y^2 \qquad \text{b) }x^2 + y^2\qquad \text{c) } e^x\qquad \text{d) }\operatorname{Im}((x + iy)^5)$$ I had this question come up ...
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1answer
21 views

Integrating $u^2(x+y)$ over $\partial B_1 (0)$, where $u$ is harmonic

Let $u : \mathbb{R}^n \longrightarrow \mathbb{R}$ be a harmonic function. I must prove that $$\int_{\partial B_1 (0)} u(x+ty) u \left( x+\frac{1}{t}y \right) dS_y = \int_{\partial B_1 (0)} u^2(x+y) ...
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The eigenvalue of Laplace problem on a domain with a line removed

Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
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1answer
9 views

Could a non-constant harmonic function be bounded or has extrema ? Could it exist in the physical world?

Harmonic function is a function which its Laplacian is equal zero: $$ {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}} =0$$ Harmonic functions have the mean value ...
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17 views

Harmonic polymonials approximating function

Can we approximate $f(x)=-x_1^2/2$ defined in the unit cube of $\mathbb{R}^2$ by a sequence of harmonic polynomials? We can find a compact subset $K$ of the unit cube such that it has empty ...
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1answer
22 views

Conditions for membership of $H^2$

I'm looking for conditions for a function $u$ defined on a bounded domain $\Omega\subset\mathbb{R}^n$ to be an element of the Sobolev space $H^2(\Omega)$. I heard the other day that if $u$ is harmonic ...
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The unique tangent of super-harmonic function on the liminf of the singularity

Let $n\ge2,B=B(0,1)\subset\mathbb{R}^n$ is the unit ball. Let $u\in C^2(B\backslash\{0\}),\Delta u\le0$ be the super-harmonic function such that $\liminf\limits_{x\to 0}u(x)=0$. Let $v,w\in C^2(B)$ ...
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45 views

Proof that Harmonic Implies Conformal

How do I show that for some function $u$ that $$\Delta u = 0 \implies u \> \> \text{is analytic}$$ and assuming $u$ has non-vanishing derivative everywhere, how do I show $u$ is conformal? ...
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The maximum principle of positive super-harmonic function

Let $0<u\in C^3(\mathbb R^n),\Delta u\le0$, Show that $\forall r>0,\forall|x|\ge r,|x|^{n-2}u(x)\ge\min\limits_{|y|=r}u(y)r^{n-2}$ Using Kelvin transform $\displaystyle v(x)=\left(\frac {r}{|x|}...
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30 views

Integrating spherical harmonic function

How do you evaluate $$\int_{0}^{2\pi} \int_{0}^{\pi} \sin \theta ~ Y_{lm}(\theta,\phi) \mathrm d\theta \mathrm d\phi $$ where $Y_{lm}(\theta, \phi)$ is the spherical harmonic defined as $$Y_{lm} (\...
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0answers
15 views

Non-uniqueness of solutions to the steady-heat equation on the disk that do not converge uniformly to the boundary

According to exercise 18, chp. 2, of Stein & Shakarchi's Fourier analysis, $\frac{\partial P_r(\theta)}{\partial \theta}$ is a solution to the steady-heat equation that converges only pointwise to ...
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1answer
54 views

Showing harmonicity of $\frac{1}{|x|}H(\frac{x}{|x|^2})$ with $H$ harmonic on $\mathbb{R}^3$.

In this year's first exam, my teacher gave the following problem: State and prove the spherical average theorem for harmonic functions in open sets of $\mathbb{R}^n$. Let $H$ be a harmonic ...
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1answer
32 views

How do I show this function is harmonic?

In an exam, my professor gave the following exercise: State and prove the mean value theorem for harmonic functions. Let $H$ be a harmonic function on $\mathbb{R}^n$. Show all of its dilations $H^\...
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1answer
33 views

For $f \in C^2 : \mathbb{R}^3 \rightarrow \mathbb{R}, s.t. ∆f > 0$ maximum of $f$ on $B(0,r)$ is a strictly increasing function of $r$.

For $f \in C^2 : \mathbb{R}^3 \rightarrow \mathbb{R}, s.t. ∆f > 0$ prove that the maximum of $f$ on $B(0,r)$ is a strictly increasing function of $r$. I can take the differtial of $f$ and use ...
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Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
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1answer
28 views

What is wrong about this proof for the mean-value theorem for harmonic functions?

Let $\Omega\subset\mathbb{R}^n$ be an open connected domain, and let $u\in C^2(\Omega)$ be a harmonic function on $\Omega$. Then for every ball $B_R(x)=\{y\in\Omega:|x-y|<R\}$ in $\Omega$ we have $...
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1answer
27 views

Potential theory, potentials and harmonic functions

In the development of potential theory we mostly study harmonic functions. However I found some paper, which present potential theory as the study of potentials. Are potentials harmonic functions?
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76 views

Trouble with an application of Green's representation formula

The teacher solved an exercise in class which required you to prove that, if $\Omega$ is a bounded domain in $\mathbb R^n$ and $G$ its Green function, then $G$ is symmetric, i.e. $G(x,y)=G(y,x)$ for ...
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1answer
21 views

square of polynomial still harmonic? [closed]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
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3answers
48 views

$u$ and $u^2$ are harmonic.

Let $D$ be the unit disk centered at $0$ in the complex plane, and let $u$ be a real harmonic function on D. Find all $u$ such that $u(0)=0$ and $u^2$ is also harmonic on $D$.
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Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
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0answers
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Removing Singularity of real bounded harmonic in punctured disk. [duplicate]

If $u(z)$ is real harmonic and bounded in the punctured disk $0<|z-z_0|<R.$ Show that $\lim_{z\to z_0} u(z)$ exists. I already know Complex analytic function $f$ which has singularity $z_0$ ...
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0answers
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Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
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An example of a bounded domain $\Omega\subset \left\{ 0<\Re s< 1\right\} $ for which $\Re \zeta(s)$ is non-negative

Denoting the complex variable $s=\sigma+it$ (and we know that $\mathbb{C}$ and $\mathbb{R}^2$ are isomorphic, thus $s\equiv(\sigma,t)\in\mathbb{R}^2$) one has for $0<\Re s=\sigma<1$ that $$\zeta(...
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2answers
70 views

Construct a harmonic function that appears to be discontinuous on the unit circle.

Construct a harmonic function $u$ in $D(0,1)$ that satisfies $$ lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \...
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0answers
41 views

Mean Value Property for harmonic functions

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
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1answer
51 views

Decomposition of Harmonic function into sum of holomorphic and anti-holomorphic function

How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\...
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1answer
31 views

Does there exist a kernel concept for Taylor expansions?

In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ...
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Harmonic Function - Multivariable calculus

One more exercise I stepped at while strolling through papers and journals for my preparation on the semester exams for multivariable calculus. Let $D=\{(x,y): x^2 + y^2 \leq 1\}$ A function $f:D \to ...
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1answer
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Is $f(x)$ constant under these conditions?

Statement Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be an function that is concave up and increasing. If $\displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0$, then $f$ is constant. It'll be easy if ...
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2answers
46 views

Show : A holomorphic function is harmonic if $\frac{\partial f}{\partial \overline{z}}=0$

Let's consider a "new" basis of the partial differential operators (of order 1) on $\mathbb{R^2}\approx\mathbb{C}$ defined by : $\frac{\partial}{\partial z}:= \frac{1}{2}(\frac{\partial}{\partial x}-...
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Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$ \Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
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1answer
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Prove: For a harmonic function $u$, $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ implies $u = 0$ for $0 < \epsilon < 0.5$

Prove: For a harmonic function $u: \mathbb{R}^3 \rightarrow \mathbb{R}$, if for all $(x,y,z) \in \mathbb{R}^3$ $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ then $u = 0$ for $0 < \...
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1answer
42 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on $...
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0answers
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Is there a mistake in this proof in Rudin's RCA?

Rudin's Real and Complex Analysis, 3rd edition, page 236 : How did Rudin get to $$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$ It seems to me that he used the mean ...
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2answers
173 views

Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion?

(First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ...
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1answer
26 views

A radial harmonic function on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$

Prove: A radial harmonic function $f$ on $\mathbb{R}^N \setminus \{0\}$ is of the form $\frac{b}{|x|^{N-2}} + c$ for $b,c \in \mathbb{R}$. My try: Label $g_i = (0,..,0,x_i,0,..,0)$. From the maximum ...
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1answer
25 views

Is it possible to analytically solve Laplace's equation between two rectangles?

I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below: Is it possible to do that analytically? So far I haven’t ...
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Non-constant harmonic function satisfying given property

Let $u(x,y)$ be a non-constant harmonic function in region $\mathbb{D}_{\mathbb{R}}=: D$ and $$A:=\{(x,y)\in D : u_x = u_y = 0\} $$ what can one say about the set $A$? Since $u$ is harmonic, there ...
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1answer
36 views

Show $u$ satisfy poisson equation

Let $f\in \mathcal S(\mathbb R^n),n\geq 3$. How can I show that $$u(x)=C_n\int_{\mathbb R^n}|x-y|^{2-n}f(y)dy$$ where $C_n$ is a suitable constant solve Poisson equation ? i.e. $$\Delta u=f.$$ The ...
3
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1answer
76 views

$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant.

$f$ is holomorphic in Ω such that $|f|^2$ is harmonic; we need to show that $f$ is constant. solution of the question In the solution attached, I don't really understand the transition between $∆|f(z)...
2
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1answer
19 views

Geometry of level sets of an harmonic function

Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open ...
0
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1answer
16 views

A question related to Laplace equation on pde.

Let $B_R(p)$ the open ball of radius R centered at p in $R^n$ and consider the following problem $∆u = 0 $ in $ R^n\setminus B¯_R(p)$ ,$u = c$ on $∂B_R(p)$, where c is a given constant. Find a non-...