# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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|}... 0answers 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} (\... 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 ... 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 ... 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^\... 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 ... 0answers 31 views ### 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 ... 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 ... 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? 0answers 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 ... 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 ? 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. 0answers 19 views ### 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 ... 0answers 12 views ### 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 ... 0answers 17 views ### 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 ... 0answers 15 views ### 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(... 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 $$lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \... 0answers 41 views ### Mean Value Property for harmonic functions I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ... 1answer 51 views ### Decomposition of Harmonic function into sum of holomorphic and anti-holomorphic function How do you prove that a harmonic planar mapping f(x,y) = u(x,y) + i v(x,y) for real u,v can be written as f(x,y) = \phi(x,y) + \overline{\psi}(x,y) where \phi is a holomorphic function, and \... 1answer 31 views ### Does there exist a kernel concept for Taylor expansions? In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels. ... 0answers 52 views ### Harmonic Function - Multivariable calculus One more exercise I stepped at while strolling through papers and journals for my preparation on the semester exams for multivariable calculus. Let D=\{(x,y): x^2 + y^2 \leq 1\} A function f:D \to ... 1answer 78 views ### Is f(x) constant under these conditions? Statement Let f:\mathbb{R}\rightarrow \mathbb{R} be an function that is concave up and increasing. If \displaystyle \lim_{x\to \infty}\frac{f(x)}{x}=0, then f is constant. It'll be easy if ... 2answers 46 views ### Show : A holomorphic function is harmonic if \frac{\partial f}{\partial \overline{z}}=0 Let's consider a "new" basis of the partial differential operators (of order 1) on \mathbb{R^2}\approx\mathbb{C} defined by : \frac{\partial}{\partial z}:= \frac{1}{2}(\frac{\partial}{\partial x}-... 0answers 32 views ### Are harmonic functions (\Delta u=0) with compact support unique? Does anyone how to solve the problem$$ \Delta u=0$$with u \in R^n and u(x) \to 0 as |x| \to \infty? Is u=0 the only solution? Many thanks. 1answer 59 views ### Prove: For a harmonic function u, |z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon} implies u = 0 for 0 < \epsilon < 0.5 Prove: For a harmonic function u: \mathbb{R}^3 \rightarrow \mathbb{R}, if for all (x,y,z) \in \mathbb{R}^3 |z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon} then u = 0 for 0 < \... 1answer 42 views ### Harmonic forms and functions on compact manifolds I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let X be a compact manifold and A^k(X) be the space of C^\infty forms on ... 0answers 72 views ### Is there a mistake in this proof in Rudin's RCA? Rudin's Real and Complex Analysis, 3rd edition, page 236 : How did Rudin get to$$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$It seems to me that he used the mean ... 2answers 173 views ### Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion? (First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ... 1answer 26 views ### A radial harmonic function on \mathbb{R}^N \setminus \{0\} is of the form \frac{b}{|x|^{N-2}} + c Prove: A radial harmonic function f on \mathbb{R}^N \setminus \{0\} is of the form \frac{b}{|x|^{N-2}} + c for b,c \in \mathbb{R}. My try: Label g_i = (0,..,0,x_i,0,..,0). From the maximum ... 1answer 25 views ### Is it possible to analytically solve Laplace's equation between two rectangles? I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below: Is it possible to do that analytically? So far I haven’t ... 2answers 60 views ### Why is \sqrt r \cos \frac \theta 2 harmonic? [closed] Why is \sqrt r \cos \frac \theta 2 harmonic? 0answers 28 views ### Non-constant harmonic function satisfying given property Let u(x,y) be a non-constant harmonic function in region \mathbb{D}_{\mathbb{R}}=: D and$$A:=\{(x,y)\in D : u_x = u_y = 0\} $$what can one say about the set A? Since u is harmonic, there ... 1answer 36 views ### Show u satisfy poisson equation Let f\in \mathcal S(\mathbb R^n),n\geq 3. How can I show that$$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 ... 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)...
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 ...
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-...