Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
229 questions
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Possible flaw in Karp and Margulis's proof of convexity of null quadrature domains?
I was reading L. Karp and A. Margulis's proof of the convexity of the complement of a null quadrature domain. This paper is cited by many others, for example, S. Eberle, A. Figalli and G. Weiss.
...
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-3
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1
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$H_{k^n} - H_{k^{n-1}-1} > H_k - \gamma$ for all $k,n$ natural numbers?
Let $H_n = \sum_{i=1}^n \frac{1}{i}$ be the $n$-th harmonic number and $\gamma$ be the Euler–Mascheroni constant. I am investigating the inequality:
$$H_{k^n} - H_{k^{n-1}-1} > H_k - \gamma$$
The ...
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2
votes
1
answer
324
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Proof of $H_{k^2} - H_{k-1} > H_k - \gamma$ for all $k$ natural number
Let $H_n = \sum_{i=1}^n \frac{1}{i}$ be the $n$-th harmonic number and $\gamma$ be the Euler–Mascheroni constant. I am investigating the inequality:
$$H_{k^2} - H_{k-1} > H_k - \gamma$$
The ...
- 273
3
votes
1
answer
411
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Existence of a harmonic function on a bounded open set in euclidean space extending a continuous function on its boundary
Let $U \subset\mathbb{R}$$n$ be an arbitrary connected bounded open set, and let $f$ : $∂U$ $\rightarrow$ $\mathbb{R}$ be continuous, where $∂U$ denotes the set $\overline{U} - U$.
Are there ...
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0
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90
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Transfer of spectral data from stratified surfaces to embedded graphs
Let $\Gamma \subset X$ be a finite connected $(q+1)$-regular Ramanujan multigraph, embedded as the 0,1-skeleta of a stratified (smooth away from $\Gamma$) 2-complex $X \subset \mathbb{R}^3$.
A ...
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$\gamma > S(k) := \sum_{m=2}^{k}\frac{(-1)^m}{m}H_{k-m+1}^{(m)}$
Proving $\gamma > S(k)$ for all $k\geq2$
Given:
$$\gamma = \sum_{m=2}^\infty (-1)^m\frac{\zeta(m)}{m}, \quad H_n^{(m)}=\sum_{j=1}^n\frac{1}{j^m}$$
Conjecture:
$$\gamma > S(k) := \sum_{m=2}^{k}\...
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2
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0
answers
95
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Sharpness of boundary decay for second derivatives of harmonic functions in the unit ball
Question:
Let $B_1 \subset \mathbb{R}^n$ ($n \geq 2$) be the unit ball, and $h$ be harmonic in $B_1$ with Dirichlet boundary data $\phi \in C^1(\partial B_1)$. Consider the scaled second derivative $(...
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1
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0
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92
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Subharmonic functions of general bilinear forms
It is well known that $|\nabla u|^2$ is subharmonic whenever $\Delta u = 0$. This is the usual Bochner identity.
I am looking for generalizations of this in the following sense: Let $u$ be a smooth ...
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3
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1
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Removable singularities harmonic functions
Let $\Sigma=\{x=(x',x'')\in \mathbb R^k\times \mathbb R^{d-k}: x'=0\}\subset \mathbb R^d$ for $2\le k \le d$.
Theorem. Suppose $u\in C^\infty(\mathbb R^d\backslash \Sigma)\cap L^\infty_\mathrm{loc}(\...
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1
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0
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uniform bound for metric tensor, its inverse and their first derivative in local coordinate on Riemannian manifold
Let $(M, g)$ be a complete, simply connected non-compact n-dimensional Riemannian manifold with pinched negative sectional curvature(i.e. its sectional curvature $-b^2 \leq K \leq -a^2 $ where $0 <...
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Can prime numbers be isolated as zeros of a harmonic wave function?
Can prime numbers be isolated as zeros of a harmonic wave function?
Equation
Prime numbers are often studied through analytic number theory, modular sieves, and spectral methods. In my recent research,...
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Long-term behaviour of the heat equation for unbounded initial data
Consider the heat equation
\begin{equation}
\partial_t u(t,x) = \Delta u(t,x)
\\
u(0,x) = f(x)
\end{equation}
on the whole space $\mathbb{R}^d$. It is for example well known that for any $f\in C_0(\...
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0
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434
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Performing an uppersemicontinuous regularization twice
I am unsure about the conclusion of a proof I wrote, the statement I am trying to prove as well as the proof itself are included below.
Exact statement I am trying to prove:
Let $f(\lambda,z): \mathbb{...
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2
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1
answer
447
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Positive harmonic functions
Let $n\geq3$ and $A\subset\mathbb{R}^n$ be a discrete subset. How to characterize all the positive harmonic function on $\mathbb{R}^n\setminus A$ ?
Rmk: This is from Exercise 17 in Chapter 3 of the ...
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2
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1
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248
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Existence of meromorphic functions on Riemann surfaces
I apologise for the slightly technical question, and I am aware of the existence of the Mathoverflow question with the same title The existence of meromorphic functions on Riemann surfaces
Proofs of ...
- 3,118
1
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0
answers
80
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Commutants of analytic Toeplitz operators on the harmonic Bergman space
It is known that, in the analytic Bergman space $L_a^2(\mathbb{D},dA)$, if the Toeplitz operator $T_{f}$ whose symbol $f$ is an analytic nonconstant function on the unit disk, commutes with $T_{u}$ ...
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2
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156
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A reference for Schauder estimates of Newtonian potential on smooth boundary
Let $n\geq 2$ and $\Omega$ be a smooth bounded open subset of $\mathbb{R}^n$. Also, let $f\in C^{m,\alpha}(\overline{\Omega})$ and define $N$ as the Newtonian potential: $$N(x,y) = \begin{cases}
...
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Is there a method to determine whether a given function on $\Bbb R^n$ is harmonic or not under any metric?
Given a function on $\Bbb R^n$, is there a method to determine whether it satisfies the Laplacian equation under some metric (may not be the Euclidean metric)?
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given a curve in a plane region, does there exist a harmonic function with maximal growth along said curve?
This question was sparked from the fact that, using the mean value property, you can bound the value of $u(1/2)$ given that $u$ is a positive harmonic function on $D(0, 1)$ and $u(0)=1$:
\begin{align*}...
8
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1
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271
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Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
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2
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0
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133
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Dirichlet problem for an elliptic operator
consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator.
My question is there is a formula of the ...
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2
votes
1
answer
279
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Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
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2
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0
answers
117
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Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space
Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
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A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
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Have strictly superharmonic functions on graphs been studied?
Given a graph $G$ and a function $f:G\to\mathbb R$, we say that $f$ is harmonic if
$$f(x)=\frac{1}{|N(x)|}\sum_{y\in N(x)}f(y)$$
for every $x\in G$, where $N(x)$ denotes the set of neighbors of $G$. ...
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0
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1
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Harmonic functions and monotonic decay
I have a general question surrounding certain harmonic functions.
I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
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2
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1
answer
314
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Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
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4
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1
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253
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Green's kernel estimates on finitely generated groups
I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
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3
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0
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129
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Upcrossing lemma and subharmonic functions
I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $
\lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
1
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0
answers
69
views
Behaviour of higher order Laplacian in punctured domain
Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
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1
vote
1
answer
248
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Carleman's Liouville theorem for entire functions bounded along every ray
There is a long history on constructing entire functions bounded along every direction. For example, we refer to Burckel's math review on Newman (Amer. Math. Monthly 1976 MR0387593) or this ...
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2
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0
answers
104
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Autocovariance of harmonic oscillator in fluid (Langevin Equation)
I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
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6
votes
2
answers
537
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$C^1$ harmonic functions on a dense open set are globally harmonic
In a paper I am studying, at a certain point the authors introduce a function $u\in C^1(B_1,\mathbb{R})$ which is harmonic in a dense open subset $U$ of $B_1$. From this, they seem to conclude that $u$...
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1
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Bounded covariant derivative of curvature tensor
Let $M$ be a complete Riemannian manifold.
Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...
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If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine
We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$
Therefore $u-u(...
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Integrability (and hence regularity) of $\alpha$-harmonic maps
To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
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An open problem of Hardy and Littlewood on $p$-integral means
In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
2
votes
1
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355
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Fourier transforms of homogeneous functions [closed]
Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P_d(x)
$$
where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
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0
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Generalized harmonic map
Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...
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3
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Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
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1
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Ratio of measure of level region for harmonic functions
Let $u$ be a harmonic function defined on $B_1(0)\subset\mathbb{R}^2$, $u(0)=0$, and $\{x\in B_1(0):u(x)>0\}$ is simply connected. Is there a universal constant $c>0$ satisfying that
$$
c\leq \...
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2
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1
answer
280
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Super harmonic function
If $u>0$ in $\mathbb{R}^n\backslash\{0\}$ ($n\geq 2$) and $-\Delta u>0$ in $\mathbb{R}^n\backslash\{0\}$, is it true that $\liminf_{|y|\rightarrow 0}u(y)>0$?
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1
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Newtonian potentials of balls and spheres
This is a simple question whose answer was probably known to Poisson, but I was not able to find it by searching. I need explicit formulas for the Newtonian potential of the unit ball $\mathbb{B}^n$ ...
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2
votes
1
answer
121
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Pair of positive harmonic functions with negative inner product in Drury-Arveson space
Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by
$$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$
Call the corresponding real reproducing kernel ...
- 1,503
2
votes
1
answer
304
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Global Hölder regularity
I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real
and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...
user124759
3
votes
1
answer
292
views
Subharmonic distributions on the plane
A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution ...
- 9,152
2
votes
0
answers
168
views
Critical points of a strictly subharmonic function
Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant:
\begin{equation}
\Delta u = A > 0....
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0
answers
158
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Estimate value of harmonic function in the annulus
Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients
$$
Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...
user124759
1
vote
0
answers
165
views
Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?
In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
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2
votes
1
answer
233
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Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
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