Questions tagged [laplacian]

The properties of the Laplace differential operator, denoted $\Delta$ or $\nabla^2$, and defined as the divergence of the gradient. For Laplace equation, see (harmonic-functions)

Filter by
Sorted by
Tagged with
1
vote
1answer
35 views

If $\Delta \varphi =0$ then $\nabla \times \nabla \varphi =0$.

I was looking into a question about vector functions $\varphi$ satisfying the Laplace equation $\Delta \varphi =0$. I found that this answer to the question was given in most places I looked for, but ...
0
votes
0answers
25 views

Multivariable Calculus, Laplacian problem

$u$ is a function of $x,y,z,$ and $t,$ but the Laplacian Equation does't involve $\frac{\partial^2 u}{\partial t^2}.$ Why is that? One explanation I have seen before is that the gradient is ...
0
votes
1answer
31 views

Eigenvalue problem on the real line

The following is a problem in a text (in Portuguese) on Critical Point Theory that I am reading: Find the eigenvalues and eigenfunctions of the problem $$ (P) \quad \begin{cases} - y'' = \lambda ...
0
votes
0answers
14 views

what is the difference between Laplacian field and a vector field? [closed]

'''code''' other than the fact that the laplacian field consists of both scalar and vector in a subset of space. is there any other difference?
-1
votes
0answers
18 views

Is it possible to find a generalized mean value equality for harmonic function on an arbitrary domain $\Omega$?

Mean value theorem of harmonic function states that for $u\in C(\overline \Omega)\cap C^2(\Omega)$, and $B(x_0,r)\subseteq \Omega$, where $\Omega \subseteq \mathbb R^n$ is connected and open, we have ...
0
votes
0answers
19 views

Derivating the expression for laplacian on a cylinder

I am trying to derive the expression of the laplacian operator in the surface of the cyilinder $C=\{(3\cos(t),3\sin(t),z):t\in[0,2pi],z\in[0,h]\}$. So I have started taking the coordinates $x=3\cos(...
1
vote
0answers
10 views

References about properties of the spectra of the Laplace-Beltrami on p-forms over homogeneous spaces

I am reading the paper "Specra and Eigenforms of the Laplacian on $\mathbb{S}^n$ by Ikeda-Taniguchi and $\mathbb{P}^n(\mathbb{C})$" and i want to know where do i can get a proof or some ideas to prove ...
0
votes
2answers
23 views

Solution to a special differential equation

I am wondering whether the following differential equation can be solved. $$\frac{\partial^{2}f}{\partial x^2}+ \frac{\partial^{2}f}{\partial y^2}+ \frac{\partial^{2}f}{\partial z^2}+ \alpha \frac{\...
1
vote
0answers
17 views

What is the interpretation of the normalized adjacency matrix raised to a power?

What the interpretation of the normalized adjacency matrix raised to a power $K$? If we take the exponent of an adjacency matrix we get the number of walks, but what about if we do that for the ...
0
votes
0answers
22 views

Solving this Laplacian Equation?

How do I solve the following? $\frac{∂^2\phi}{∂x^2}$ + $\frac{∂^2\phi}{∂y^2}$ + $\lambda^2\phi$ = $0$ To start of with, is it right to consider using the method of separation of variables, or is ...
0
votes
0answers
21 views

What is the mathematical significance of the integral of the normal derivative?

I found from Advanced Engineering Mathematics by Kreyszig that for the region $R$ and a scalar function $w(x, y)$ the following holds: $$\iint_R \nabla^2 w \ dxdy = \oint _{\partial R} \frac{\...
0
votes
1answer
19 views

Fractional Laplacian maps positive functions into positive functions?

Assume $f \geq 0$ is $C^\infty$ with compact support. Is it true that $$(-\Delta)^\alpha f \geq 0$$ where $\alpha < 1$? I tried to use some of the possible definition of fractional laplacian, ...
2
votes
0answers
18 views

Regularity of the one dimensional Poisson equation

Let $-\infty < a < b < \infty$ and set $U = (a,b)$. A weak solution of the Poisson's equation $\Delta u = f$ subject to $u = 0$ on the boundary with $f \in L^2(U)$, is a function $u \in H_0^1(...
2
votes
1answer
44 views

Green's Function for Dirichlet problems

I have been studying Green's functions for Laplace/Poisson's equation and have been having some trouble on a few things. In Strauss's book he claims the solution to the Dirichlet problem is: $$u(\bf ...
1
vote
1answer
28 views

Generalization of Gradient Using Jacobian, Hessian, Wronskian, and Laplacian?

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ...
0
votes
0answers
11 views

prove that a solution of Laplace equation is correct

Prove that $$U(x,y,z)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}A(k_1,k_2)exp(i(xk_1+yk_2)-zk_3)d{k_2}d{k_1}$$ is a solution for the Laplace equation in cartesian coordiantion by plugging it into ...
0
votes
1answer
13 views

Definition of generalized laplacian by Green's theorem

I am reading a book on Potential Theory and they motivate the generalized Laplacian by saying that Green's theorem implies the following: $$ \int_D \phi \Delta u = \int_D u \Delta \phi dA $$ where $\...
0
votes
0answers
26 views

Laplace-Beltrami Operator in terms of the Hodge-$\star$-Operator and the Codiferential

Given a riemannian Mainifold $(M,g)$, the Hodge-$\star$-Operator, the codifferential as $$\delta:\Omega^k(M)\rightarrow\Omega^{k-1}(M):\omega\mapsto(-1)^{n(k-1)-1}\star d\star \omega$$ and the ...
1
vote
2answers
31 views

In Laplace's equation, why is it that there is only one solution for a particular boundary value?

I know that there is a unique solution to Laplace's equation that has particular boundary value. But I do not understand why this is the case. Thanks for your help in advance! PS: this is a follow-up ...
0
votes
0answers
45 views

Is this function identically zero??

Let D be a bounded domain in R^n. Show that the problem ∆v−v^3 =0 in D; v=0 on ∂D has no solution other than v ≡ 0. I am thinking I need to prove it to be harmonic in order to apply Green's or ...
0
votes
1answer
18 views

How are eigenvalues and eigenfunctors of operators like Laplacian understood?

Normally you would consider eigenvectors only of operators from the same space to itself. But the Laplacian is usually defined on $C^2$ into $C$, which is a supset of $C^2$. Which leads to my ...
0
votes
1answer
21 views

Scalar fields proof

How to prove that for every two scalar fields $u(x,y,z)$ and $v(x,y,z)$ the identity above holds true? I guess it says the Laplacian of the dot product of two scalar fields equals the Laplacian of ...
1
vote
1answer
38 views

Voltage Inside a Spherical Shell

The question I am trying to solve involves calculating the voltage function $V(r, \theta)$ ($\theta$ is the zenith angle) inside a spherical shell with radius $R$, where the voltage along the ...
0
votes
0answers
16 views

Laplacian in polar coordinates for constant r

So, this is a problem actually motivated by a physics problem that I was facing but is again a math question. So, in polar coordinates, we can write $x=r\cos(\theta)$ and $y=r\sin(\theta)$ and using ...
1
vote
1answer
45 views

Brownian mottion and hitting probabilities

Let $D$ be a domain in $\mathbb{R}^d$ and $A$ a measurable subset of its boundary $\partial D$. For $x \in D$ define $$\phi (x) = \mathbb{P}(X_T\in A) $$ where $(X_t)$ is a Brownian Motion in $\...
1
vote
0answers
23 views

Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$ [migrated]

Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^d)$ monotone? i....
2
votes
0answers
30 views

Construction of purely radial harmonic function

So my question is In a spherical shell with $1<r<2$ construct a purely radial harmonic function v such that it takes the values $5$ and $4$ at $r=1$ and $r=2$ , respectively I know that I ...
0
votes
1answer
25 views

Problem of separation of variables for Dirichlet boundary data of Laplace's equation in polar coordinates

Need help here with figuring out boundary conditions for this problem. Also, for (i), I do know a general way or method but here I am confused since from both equations how do I find out my desired ...
0
votes
1answer
19 views

Find a bounded but non-constant solution of Poisson's equation

Consider Poisson's equation with homogenous Dirichlet boundary conditions \begin{equation} \begin{split} \Delta u&=g \text{ in } B_R(0)\\ u&=0 \text{ on } \partial B_R(0), \end{split} \...
0
votes
0answers
12 views

Vortex solution of Laplace equation (XY model)

The hamiltonian of XY model, which is closely connected with BKT - transition is following: \begin{equation} H=\frac{J}{2} \int \text{d}^2 r \, \nabla \varphi \cdot \nabla \varphi, \quad \...
0
votes
0answers
37 views

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 a externally heated axi-symmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
2
votes
0answers
29 views

Solving $\Delta u=f(u)$

I am currently reading Partial Differential Equations on Riemannian manifold without boundary. I want to consider a PDE of the form $$\Delta u=f(u).$$ Can anyone tell me a reference which contains ...
0
votes
0answers
21 views

Laplacian of Kelvin Transform - Confusion about notation

There is an exercise in Evans' book (chapter 2, problem #11). Suppose $u:\mathbb{R}^n\to\mathbb{R}$ is harmonic. Show that $\overline{u}(x)=u(\overline{x})|\overline{x}|^{n-2}=u(x/|x|^2)|x|^{2-n}$ is ...
2
votes
1answer
57 views

Definition of Tensor Laplacian

Let $(M,g)$ be an orientable Riemannian manifold, $\nabla$ its Levi-Civita connection and $\epsilon$ its volume form. Let $f\in C^\infty(M)$ be a scalar field. Then we know that we can define its ...
1
vote
1answer
28 views

A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
1
vote
2answers
58 views

Rotational invariance of Laplacian operator

I was reading in Wikipedia about Rotational invariance and noticed that the two-dimensional Laplacian operator $\nabla^2 = \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}$ is ...
0
votes
0answers
13 views

Are there any bounds on the decay rate of the spectrum of graph laplacians?

I am trying to analyze graph laplacians and in that context, I want to know about the spectral decay of graph laplacians i.e. how fast does the tail sum of its eigenvalues decays? Any research answers/...
0
votes
1answer
48 views

Find all functions $g$ such that $\nabla^2g = x^3+y^2+z$

Find all functions $g$ such that $$\nabla^2g = x^3+y^2+z$$ What I tried: Basically $\nabla^2$ gradient of a divergence and we can write it as $$\nabla^2 g = \frac{\partial^2g}{\partial x^2} + \...
0
votes
0answers
16 views

integral identity with laplacian squared

I found this exercise but I am not able to solve it: let $n > 1$ a positive integer .Prove that for any $u \; \colon \, \mathbb{R}^n \to \mathbb{R}$ with $ u \in C^2(\mathbb{R}^n)$ with compact ...
0
votes
0answers
27 views

Second-order partial derivatives and Laplacian of p-norm

Are there neat formulas for second-order partial derivatives and Laplacian of the p-norm? $$ | x |_p = \left(\sum_i | x_i |^p\right)^{1/p}, \qquad p>1,\ x\in\mathbb R^n $$ I found the following: $...
0
votes
0answers
18 views

Relation between expected values of eigenvalues of Laplacian matrix of a graph and eigenvalues of expected Laplacian matrix of that graph?

Particularly, I am dealing with Erdős–Rényi random $G(n,p)$, so the expected Laplacian matrix of $G(n,p)$ is $p(J_n-I_n)$, where $J_n$ and $I_n$ are one and identity matrices, respectively. In ...
3
votes
0answers
15 views

Immersion $H^s(\Omega) \hookrightarrow H^{s'}(\Omega)$ but with fractional laplacian defined on $\mathbb{R}^N$?

I know that it holds this embeding $$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$ for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined ...
2
votes
1answer
43 views

How to show the Laplacian is a self-adjoint linear operator

I'm trying to prove the following: $$\iint_R(\phi_1\nabla^2\phi_2-\phi_2\nabla^2\phi_1)\ dR = 0$$ For $ 0 < x < L, 0 < y < H$. Given $\phi_i(x,y) $ satisfying the boundary conditions $$ \...
0
votes
0answers
26 views

Where can I find a proof of the fact that the Gaussian semigroup is generated by the Dirichlet Laplacian?

Let $H_0$ denote the Dirichlet Laplacian on $\mathbb{R}^d$ given by $H_0u=\Delta u$ for $u \in \text{dom}(H_0):= \{f \in H^1(\mathbb{R}^d): \Delta f \in L_2(\mathbb{R}^d) \}$. Further, for $t>0$ ...
3
votes
1answer
70 views

Spectrum of the bi-Laplacian

I have the following operator: $$\begin{align*}D(A)&=\{f \in C[0,1]:f''''\in C[0,1] \text{ and } f^{(k)}(0)=0 \ \forall\ 0\leq k\leq 3\}\\Af&=-f''''.\end{align*}$$ I'm trying to find the ...
0
votes
1answer
18 views

Singular point at Neumann Boundary Condition

I am trying to solve the following Boundary Value Problem - a particle is moving on the surface of a sphere: $$ \triangle y(\theta)=\frac{d²y(\theta)}{d \theta²} + \cot{\theta } \frac{dy(\theta)}{d\...
1
vote
0answers
24 views

Poisson equation on $M\times \mathbb{R}$ with exponential asymptotic condition

Let $M$ be a closed Riemannian manifold, $g$ be a smooth function with compact support on $M\times\mathbb{R}$. Equip $M\times \mathbb{R}$ with the standard product metric. My problem : Determine ...
0
votes
1answer
28 views

Laplacian of complex function proof

If $f \in C^2$(as a two variable real function) and $\Delta f = \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$ then $$\Delta f = 4 \frac{\partial^2f}{\partial z \partial \overline{...
0
votes
0answers
22 views

Green's function of a Laplacian on a Torus does not exist?

Green's function of a Laplacian on a Torus does not exist? My answer is No, it does not exist. Appreciate your help If I am wrong.
0
votes
0answers
18 views

Integration by parts Fractional Laplacian

Define the Fractional Laplacian of a Schwartz class funtion $u$ for $\lambda \in (0,1)$ as $$\mathscr{L}_\lambda[u](x):= PV \int_{|z|>0} \frac{u(x)-u(x+z)}{|z|^{d+2\lambda}}\,dz$$ I wish to prove ...

1
2 3 4 5
16