All Questions
Tagged with ap.analysis-of-pdes harmonic-functions
64 questions
25
votes
1
answer
2k
views
The origin of Discrete `Liouville's theorem'
It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...
- 12.3k
14
votes
0
answers
632
views
Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
- 6,741
11
votes
2
answers
2k
views
Harmonic function properties on $\mathbb R^3$
Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...
- 211
8
votes
3
answers
701
views
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 $V(z) \in C^{\infty}(\partial \Omega)$?
...
- 1,350
8
votes
2
answers
773
views
Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,135
8
votes
2
answers
622
views
Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 903
8
votes
2
answers
471
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a ...
- 6,741
7
votes
2
answers
2k
views
Eigenvalues of Laplace-Beltrami on half sphere
Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
- 1,385
7
votes
1
answer
977
views
Kernel of the Laplacian + a function
It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
- 369
7
votes
1
answer
339
views
Does the pointwise mean value property imply harmonicity?
Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
- 1,101
7
votes
1
answer
410
views
Limit of zero sets of harmonic functions
Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...
- 73
7
votes
0
answers
290
views
Harmonic map heat flow in positive curvature
Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
- 705
6
votes
0
answers
326
views
Are the two-valued homogeneous harmonic functions classified?
Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$?
For reference, multi-valued functions are familiar objects in ...
- 5,038
5
votes
1
answer
162
views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
- 6,741
5
votes
1
answer
342
views
harmonic extension of a curve by different parametrization
Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
- 914
5
votes
1
answer
348
views
A differential inequality involving gradient and laplacian
Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
- 311
5
votes
1
answer
255
views
Harmonic analysis, compute that this integral tends to $0$
We have the following setting.
$U$ is a bounded Lipschitz domain in the complex plane.
Consider the following classical Dirichlet problem for the Laplace operator:
$$\begin{align}
\Delta{}u&=0 \...
- 355
5
votes
1
answer
183
views
Can harmonic maps with immersive boundary conditions have singular points?
Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
- 6,741
5
votes
0
answers
545
views
Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to
\begin{equation}
\begin{cases}
-\Delta u=0 \quad &\mbox{in $\Omega$}\\
\frac{\partial ...
- 1,350
4
votes
1
answer
346
views
Is there a harmonic function with just one singular point?
Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy
\begin{equation}
\begin{cases}
\Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\
\frac{...
- 5,038
4
votes
1
answer
221
views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
- 4,135
4
votes
0
answers
243
views
How to use blow-up to prove the boundary regularity for a harmonic function
While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem:
Thm. 2.30.
Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
- 421
4
votes
0
answers
169
views
Can the rank of harmonic maps decrease far from the boundary?
Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
- 6,741
3
votes
1
answer
785
views
on an inequality of Brezis-Lieb
In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
- 3,388
3
votes
1
answer
206
views
About the proof of higher regularity boundary Harnack inequality
I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.
In the paper they used the Hopf lemma to show that $u_\nu>c&...
- 421
3
votes
1
answer
109
views
A harmonic function degenerate in one direction
Question. Let $u: B^3 \to \mathbf{R}$ be a harmonic function with $u(0) = 0$, $Du(0) = 0$, where its homogeneous harmonic blow-up is a polynomial $p = p(x,y)$ in two variables, so independent of $z$; ...
- 5,038
3
votes
1
answer
570
views
Extending a harmonic function in a ball to subharmonic in a larger ball
Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad \...
- 632
3
votes
1
answer
541
views
regularity of p-harmonic functions
We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.
Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
- 41
3
votes
1
answer
273
views
Harmonic interpolation with analytic initial condition
Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold.
Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function.
Is there a Harmonic ...
- 356
3
votes
0
answers
85
views
Differentiability of a weak solution
Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
- 721
3
votes
0
answers
117
views
Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
- 6,741
3
votes
0
answers
156
views
Wolff's article: Note on counterexamples in strong unique continuation problems
I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, ...
- 751
3
votes
0
answers
169
views
Why a cone/parabolic set for the nontangential maximal function?
Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...
- 81
2
votes
1
answer
520
views
The flow of Harmonic vector fields
A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.
Motivated by conversations on this questions we ask:
...
- 356
2
votes
1
answer
203
views
Reference for harmonic functions in cylinders
Question. What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders?
I am trying to gain a better understanding of the behavior of ...
- 5,038
2
votes
1
answer
203
views
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 $...
- 375
2
votes
1
answer
106
views
'Dirichlet problem' along axis for harmonic functions
Question. Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the ...
- 5,038
2
votes
1
answer
184
views
Positive subharmonic functions with constant integral blowing up at boundary
Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying
$\Delta f_n\ge 0$ (subharmonic)
$f_n\ge 0$
$\int_\Omega f_n=I>0$ ...
- 287
2
votes
1
answer
170
views
The positive solutions of the weighted Laplacian equation
Let $u$ be a positive function on $\mathbb R^n$ such that
$$
\Delta u-\partial_{x_1}u=0,
$$
where $\Delta$ is the Laplacian operator $\partial_{x_1}^2+\partial_{x_2}^2+\cdots+\partial_{x_n}^2$.
Can ...
- 2,535
2
votes
1
answer
295
views
Laplace equation in a domain with holes
Suppose $u_r(x_1,x_2)$ in $B_1(0) \setminus B_r(0)$ satisfy
\begin{align*}
\begin{cases}
\Delta u_r &=0\\
u_r(|x|=r)&=-\log r \\
u_r(|x|=1)&=0
\end{cases}
\end{align*}
with $0<r<1$....
- 54
2
votes
1
answer
197
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
- 43
2
votes
0
answers
85
views
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 ...
- 21
2
votes
0
answers
86
views
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 "...
- 1,407
2
votes
0
answers
134
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....
- 5,038
2
votes
0
answers
53
views
Has the nodal map been studied?
Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary.
Define ...
- 5,038
2
votes
0
answers
147
views
Dimension of critical set of p-harmonic function
Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...
- 91
2
votes
0
answers
113
views
$W^{1,p}-$regularity on the boundary for solution of Laplace equation with Robin boundary condition
I came across with the attached paper and here is the part that I try to understand.
If the non-tangential maximal function of $\nabla u$, i.e $(\nabla u)^*$, belongs in $L^p(\partial \Omega)$, then ...
- 227
2
votes
0
answers
104
views
Existence of harmonic maps between loops
Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
$$E[\...
- 9,853
1
vote
1
answer
200
views
Smooth approximation of a subharmonic function in the barrier sense
Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f_{\epsilon}$ ...
- 2,535
1
vote
1
answer
350
views
Functions orthogonal to harmonic functions
Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose
$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
