# 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)

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### help me understand the Spectral theorem for the laplacian

Let $u:Ω→R$ be the solution of: $∆u=λu$ and $u=0$ on $∂Ω$ Let $S=$ The spectrum of $∆ =$ all the values of $λ$ for which there is a solution. If I understand correctly if $Ω$ is bounded, we have the ...
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### Laplace equation in 2D with all 0 boundary conditions

I have a pretty basic questions for which I can't find the answer elsewhere. Suppose we have a 2D Laplace equation $$\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}=0$$ ...
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### Invariance of the Laplacian under orthogonal transformations

I am struggling to solve the following question. Consider the function $u(x, y)$ which is twice-differentiable in its arguments and in arguments $\xi$ and $\eta$ obtained by the following linear ...
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### Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
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### Hyperbolic paraboloids as solutions to Laplace's equation

I decided to try my hand at solving Laplace's equations in as many ways as I could, and I've come across the result that non-rotated conic sections (those which do not depend on $xy$ arise naturally ...
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### Boundedness of the Laplacian Eigenfunctions

I have a doubt regarding the Laplacian eigenfunctions $\left\{\phi_n\right\}_{n=1}^\infty$ with Dirichlet boundary conditions. I know that the functions form an orthonormal basis in $L^2$ and an ...
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### Correlation Between Laplacian And Convexity Around Point In R^D Euclidean Space [closed]

So, in 1D Euclidean space, the Laplacian of a function is simply the second derivative of that function. In this case, the sign of the Laplacian at extremum points will tell us the convexity of that ...
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### Is $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ by itself a valid operator?

If I simply consider just one of the combinations $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ both of them take from $\Omega^r(M)\to \Omega^r(M)$ for some manifold $M$. But do they ...
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### Analytical Solution of a Laplace Equation with Given Boundary and Initial Conditions

I'm trying to solve the following Laplace equation analytically: \begin{align*} \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} &= 0 \end{align*} Subject to the boundary ...
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### Dimensional analysis of Laplacian

Given a function $$f(x, y): \mathbb{R} \left[kg \right] \times \mathbb{R} \left[K \right] \mapsto \mathbb{R} \left[m \right]$$ where the units of the variables $x, y$ and of the function $f(x, y)$ are ...
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### Terminology: functions proportional to own Laplacian

Is there a generic term for functions that are proportional to their own Laplacians, and/or for basis sets composed of such functions? Simple examples would include the well-known Fourier ...
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### Why does my solution to $\Delta u = \lambda u$ contradict the regularity theorem?
I am confused about the regularity theorem for Laplacian. It states that if we take a weak solutions of $\Delta u = \lambda u$, then $u$ must be a smooth function. But I cannot understand this ...