All Questions
Tagged with laplacian ap.analysis-of-pdes
71 questions
1
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1
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59
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Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains
On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$.
I'm interested ...
- 153
1
vote
0
answers
89
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Euler-Lagrange equation of fractional Laplacian
The following result is in "An extension problem related to the fractional Laplacian" Section 3.2 by Caffarelli-Silvestre. I’m confused how to show it and wish to have some help.
Suppose $u:\...
- 421
0
votes
0
answers
40
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Is the principal eigenvalue of the clamped plate minimized by the equilateral triangle among all triangles of a given area?
Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$
(bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by
$$\Delta^2 u = \lambda u $$
$$ u|_{\partial \...
- 3,245
1
vote
0
answers
88
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Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
3
votes
0
answers
153
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Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
1
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0
answers
43
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Behaviour of higher order Laplacian in punctured domain
Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
- 11
4
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1
answer
147
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Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
- 41
3
votes
0
answers
82
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Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
- 2,095
2
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1
answer
639
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Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
- 721
7
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0
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123
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Steklov eigenvalue for circle valued functions
Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization:
$$\sigma_1(M,g)...
- 2,095
2
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0
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141
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A question about Gauss-Green formula - a weaker assumption
The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place
$$\...
- 1,759
2
votes
1
answer
118
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Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
- 151
1
vote
0
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101
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Laplace equation with bisection boundary condition on upper side
solve Laplace equation $ U_{xx} + U_{yy} = 0 $ with given boundary
condition : $ (x,y) \in (0,a) \times (0,b) $ $\begin{cases} U_x(0,y)
= U(x,0) = U_x(a,y) = 0 \\ U(x,b)= c= constant \hspace{3mm} ,...
0
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0
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136
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Fractional Laplacian of smooth cut off functions
Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$
...
- 637
1
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0
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239
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Asymptotics for the first eigenvalue for the Laplace-Beltrami operator on the sphere
I am trying to understand the existence of positive solutions for the following equations,
$-\Delta_{\mathbb{S}^n} u + \lambda u = f(u)$
where $f$ is some non-linearity, say $f(t)=t^3.$
By considering ...
- 637
7
votes
1
answer
1k
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Eigenvalues and eigenfunctions of the Laplace operator on entire plane
According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
- 597
1
vote
0
answers
119
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Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls
Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
- 201
0
votes
1
answer
121
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Inequality involving the fractional Laplacian
Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
- 205
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1
answer
541
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Do Laplace-Beltrami eigenfunctions vary continuously with the metric?
I'm interested in Laplace Beltrami operators $$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $...
- 113
0
votes
0
answers
131
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Green kernel vs fundamental solution
Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...
- 650
1
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0
answers
84
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How to show that $\Delta W \leq −2(n − 4)V$?
I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below,
...
- 637
2
votes
1
answer
225
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Property about the fractional Laplacian
Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
- 205
4
votes
1
answer
377
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Differential inequalities under which a flat function must be identically zero
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $.
Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
- 356
2
votes
0
answers
52
views
Deducing elliptic regularity using spherical formula for Laplacian
It is well known that if $\Delta u = f$ and $u\in H_0^1(U),f \in L^2(U)$ for some domain $U,$ then we have $\|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}$ for some constant $C.$ One way to prove this is to ...
- 1,755
3
votes
1
answer
402
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Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
- 1,759
3
votes
1
answer
80
views
Estimate of the norm of the radial part of a function
Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
- 1,755
2
votes
0
answers
206
views
Laplacian on a manifold with two boundary components
I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
- 207
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0
answers
70
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Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
- 47
4
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0
answers
170
views
Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user39481
6
votes
2
answers
2k
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The spectrum of the Hodge Laplacian on a Riemannian manifold
The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
- 7,736
1
vote
1
answer
807
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Maximum principle of fractional Laplacian
Suppose $u$ is a sign changing classical solution of the fractional Laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega;
u=g \text{ in } \mathbb R^N -\Omega .$$
(a)Is it true that ...
- 407
6
votes
2
answers
624
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
- 356
3
votes
1
answer
171
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Spectrum of the Laplacian on the quotient of $3$-sphere
Given a finite subgroup $\Gamma$ of $O(4)$ acting freely on $S^3$, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric $2$-tensor on $S^3/\Gamma$ equipped with ...
- 2,535
5
votes
1
answer
457
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An alternative representation of the principal symbol of the Laplace operator
Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure.
Are the following two conditions equivalent?
First condition ...
- 356
2
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0
answers
112
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May the heat kernel of a connection Laplacian vanish?
Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
- 5,407
5
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0
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132
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Laplace Beltrami eigenvalues on surface of polytopes
The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
- 3,209
6
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0
answers
88
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Density of squares of radial eigenfunctions
The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
- 8,086
3
votes
0
answers
403
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Asymptotic behavior of the Dirichlet-Laplacian eigenvalues [closed]
I found in a math book http://www.cambridge.org/dz/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-partial-differential-equations?format=PB&...
- 617
3
votes
0
answers
87
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Estimate a function given an estimate of its Laplacian
Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions:
\begin{eqnarray*}
\int_B |f_\lambda(x)|^2dx\leq 1,...
- 21.8k
10
votes
3
answers
541
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Curvature of the boundary vs. normal derivative of the first eigenfunction
Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.
Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
- 201
1
vote
0
answers
78
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For what potentials is the heat operator with a potential term hypoelliptic?
If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat ...
- 5,407
0
votes
0
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258
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Boundary regularity of the solution of a Poisson equation in a polyhedron
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
- 167
4
votes
1
answer
135
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"Designing" Nodal sets of Laplacians in 2 or 3 dimensional domains
The properties of nodal sets (i.e. zero level sets of eigenfunctions) for the first non-trivial eigenfunction for Laplacians have been studied extensively.
My rough understanding is that one could ...
- 41
5
votes
1
answer
232
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certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons
For a given polygon $P_N$, with side lengths $x_1,\cdots,x_N$ and interior angles $\theta_1,\cdots,\theta_N$ let $\lambda(x_1,\cdots,x_N,\theta_1,\cdots,\theta_N)$ denote the least eigenvalue of ...
- 1,583
2
votes
1
answer
104
views
Universal constant for reverse inequality between first eigenvalues of Neumann and Dirichlet problems
I finally decided to post the following naive question but will if consensus is that it is out of the scope of this site , it will be immediately deleted.
Suppose $\Omega\subset\mathbb R^2$ is a ...
- 1,583
2
votes
0
answers
149
views
Inequalities concerning principal eigenvalues of Laplacian with different boundary conditions
Suppose $\Omega\subset\mathbb R^2$ is a bounded simply connected domain with sufficiently smooth boundary. Consider the following three BVPs (respectively with Dirchlet, Neumann and certain non-local ...
- 1,583
5
votes
0
answers
159
views
A conjecture on shape optimization for Dirichlet-Laplacian
For a given domain $G$, with sufficiently smooth boundary, in the plane we denote the first two eigenvalues of Dirichlet-Laplacian on $G$ of by $\lambda_1(G)$ and $\lambda_2(G)$.
$\textbf{Open(?) ...
- 1,583
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
1
vote
0
answers
249
views
An H2 estimate for Helmholtz equation
How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...
- 41
3
votes
1
answer
2k
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The inverse of Laplacian operator for different orders
I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you!
Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded ...
- 679