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Let us approximate the "Laplacian" (second derivative operator) in the interval $[0,1]$ using graph Laplacians on "chain graphs" with vertices $\tfrac in$, $i=0,1,\ldots,n$. … Then graph Laplacian applied to a function $f$, normalised by a factor $n^{-2}$, converges to $f''$, except at the boundary. …

On the probability side, this operator was extensively studied by Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček. I always thought that Harnack inequality was proved in already in their f …

Yes, the result is true, in arbitrary dimension $d$. Once I needed this and I could not find a reference; the following is the relevant part of the proof of Proposition 2.1 in my paper Ten equivalent …

Bogdan, The boundary Harnack principle for the fractional Laplacian, Stud. Math., 123(1) (1997), 43–80, available at http://matwbn.icm.edu.pl/ksiazki/sm/sm123/sm12313.pdf. [2] M. …

Sure one can! Just note that $f$ can be written as the "convolution" of $-g$ with an appropriate integrable kernel, the Green function $G(x)$ for the flat torus: $$f(x) = -\int_\Omega g(y) G(x - y) dy …

Indeed: the eigenvalue of the periodic $N\times N$ lattice with one vertex removed (and zero "boundary" condition a this vertex) should be approximately equal to the first eigenvalue of the usual Laplacian …

This is more of a comment, but way too long. First, two remarks on the initial part of the question: The Kelvin transform of a harmonic polynomial is of course harmonic, but it is not a polynomial. F …

Just a short addendum: long ago Stefan Samko observed this connection between Euclidean and spherical fractional Laplacians, see item d) in page 360 of his conference paper from 2000.

This means that $\sin x$ is in the domain of $(-\Delta)^s$ for any $s > 0$ (I understand that $(-\Delta)^{s/2}$ is a power of the Dirichlet Laplacian), but it does not belong to $H^s_0(\Omega)$ for any …

If true, this would follow from the integral expression for the fractional Laplacian: $$(-\Delta)^{s/2} f(x) = \int_{-\pi}^\pi (f(x) - f(y)) \nu(x - y) dy$$ for an appropriate kernel $\nu$. …

Edit: I was thinking about $(-\Delta)^{\alpha/2}$ rather than $(-\Delta)^\alpha$, so the $\alpha$ in the following answer is equal to $2\alpha$ with the notation of the statement of the question. As s …

I do not there is a strong relation between the two notions in the general case. Curvature is obviously a local object. On the other hand, the behaviour of the first Dirichlet eigenfunction near a bou …

For the record, the result requested in the question is given in Theorem 1.3.3 in: E.B. Davies, Heat Kernels and Spectral Theory, DOI:10.1017/CBO9780511566158. The assumptions are: $L$ is a positive …