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Tagged with laplacian green-function
7 questions
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Green's function of the conformal Laplacian
I am reading T. Parker, S. Rosenberg, "Invariants of conformal Laplacians", J. Differential Geom. 25(2): 199-222 (1987). I would like to understand how Green function changes if the metric ...
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The existence of a positive Green function for the Laplacian on $\mathbb R$
One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
- 5,407
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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
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Green's Function for Fractional Laplacian on the Union of Two Balls
I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:
$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
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hyperbolic "Green function" on a product of upper half-planes
Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...
- 7,609
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Does the green kernel converge as a series of functions?
Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...
user50806
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Existence of Green's function and the Dirichlet problem
Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \...
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