All Questions
Tagged with harmonic-functions elliptic-pde
32 questions
0
votes
0
answers
146
views
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 $c=0$, ...
- 169
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
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
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
0
votes
0
answers
120
views
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|...
- 375
4
votes
1
answer
3k
views
An inequality for harmonic functions
In a paper that I am reading the author quotes the following result about harmonic functions. According to him this should be "easy to show" but I don't seem to be able to do so.
Let $u:\...
- 1,149
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
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
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
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
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
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
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
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
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
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
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
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
0
votes
0
answers
184
views
Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$.
Consider the Laplace equation with a Neumann boundary condition
$$
-\Delta u = 0 \quad\mbox{in } \Omega, \qquad
\frac{\...
- 11
5
votes
1
answer
183
views
Can harmonic maps with immersive boundary conditions have singular points?
Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
- 6,741
2
votes
1
answer
170
views
The positive solutions of the weighted Laplacian equation
Let $u$ be a positive function on $\mathbb R^n$ such that
$$
\Delta u-\partial_{x_1}u=0,
$$
where $\Delta$ is the Laplacian operator $\partial_{x_1}^2+\partial_{x_2}^2+\cdots+\partial_{x_n}^2$.
Can ...
- 2,535
4
votes
0
answers
169
views
Can the rank of harmonic maps decrease far from the boundary?
Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
- 6,741
5
votes
1
answer
162
views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
- 6,741
1
vote
0
answers
145
views
On the solution of Laplace equation with mixed boundary condition
Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system
$$
\begin{...
- 363
4
votes
1
answer
166
views
A question about harmonic function
Let $u$ be a harmonic function inside the unit ball $B(0, 1)$ in $\mathbb{R}^2$ so that $|u|\leqslant 1$. Does a function u which satisfies $|\nabla u(0)|>1$ exist? If not, please prove that $|\...
- 63
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
0
votes
1
answer
390
views
Harmonic/Subharmonic lifting of functions on an annulus
Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...
0
votes
1
answer
67
views
Oblique derivative smoothness of harmonic functions
Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary
is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
- 2,715
0
votes
1
answer
497
views
Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$
I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...
- 632
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
151
views
Extendability of $L^{p}$ harmonic functions
Let $u$ be a harmonic function on some open set $\Omega\subset\mathbb{R}^{n}$ and $u\in L^{p}\left(\Omega\right)$. Is there any reference on extending $u$ to harmonic function on a larger open set $\...
- 325
1
vote
1
answer
260
views
A proof of energy functional appearing in the regularity of elliptic and parabolic equations
I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
- 11