All Questions
Tagged with ap.analysis-of-pdes harmonic-functions
64 questions
1
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3
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258
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Harmonic Function with special property
I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...
- 4,135
1
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1
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400
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The reproducing kernel for harmonics on compact manifolds
Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
- 617
1
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1
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251
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Is maximum principle valid in the case of non-smooth boundaries?
Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth
or Lipschitz, they may be very bad.
Denote $U=U_2 \...
- 11
1
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1
answer
126
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Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field
Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
- 356
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0
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43
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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 $\Delta u=0$ ...
- 11
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0
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117
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Mean Value Inequality with Linear Term
I am having trouble proving this modified mean-value inequality.
Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$
Prove that there exists constants $r_0,C>0$ depending only on $c$...
- 11
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0
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105
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Is every "higher-order" harmonic morphism conformal?
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...
- 6,741
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0
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145
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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
$$
\begin{...
- 363
1
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0
answers
192
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The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
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votes
1
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172
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Harmonic functions and monotonic decay
I have a general question surrounding certain harmonic functions.
I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
- 11
0
votes
1
answer
497
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Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$
I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...
- 632
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votes
1
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468
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Harmonic functions in infinite domain in Euclidean space
EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
- 21.8k
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184
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Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$.
Consider the Laplace equation with a Neumann boundary condition
$$
-\Delta u = 0 \quad\mbox{in } \Omega, \qquad
\frac{\...
- 11
0
votes
1
answer
390
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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(\...
