An Extension Problem Related to the Fractional Laplacian
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Additional excerpts
...See [1] for a general discussion....
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Cites background or methods from "An Extension Problem Related to the..."
...Use that the function r cos(θ/2) is a solution in the half-plane {y > 0} to the extension problem [8], div(y1−2s∇u) = 0 in {y > 0}, and that its trace on y = 0 is φ0....
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...1 is well-known (see [7]), but for the sake of completeness we sketch here a proof that uses the Caffarelli-Silvestre extension problem [8]....
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Cites methods from "An Extension Problem Related to the..."
...This approach was pointed out by Caffarelli and Silvestre in [19]; see, in particular, Section 3....
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References
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"An Extension Problem Related to the..." refers methods in this paper
...It can actually be proved using direct classical potential methods like in Landkof (1972)....
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...It can actually be proved using direct classical potential methods like in [ 9 ]....
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"An Extension Problem Related to the..." refers background in this paper
...A boundary Harnack estimate for the fractional Laplacian was first proved in Bogdan (1997) using potential methods....
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"An Extension Problem Related to the..." refers background or methods in this paper
...What we want to do is to find a map that straightens up the domain in order to apply the result of Fabes et al. (1983). First, since is a Lipschitz domain, we can find a bilipschitz map 1 n → n such that 1 x0 = 0 and 1 ∩ B1/2 = B1/2 ∩ x1 > 0 ....
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...We can then apply the result of Fabes et al. (1982b) to obtain Harnack inequality for u and thus also for f in half of the ball....
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...…derive the Harnack and boundary Harnack inequality for the fractional Laplacian from the Harnack inequality for singular elliptic equations, either with A2 weights (see Fabes et al., 1982b, 1983; Smith, 1982/83) or for certain classes of nondivergence problems (see Caffarelli and Gutierrez, 1997)....
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...This kind of problems and general properties of elliptic equations with A2 weights are studied in Fabes et al. (1982b)....
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...In the proof we can see that what we need to apply the result of Fabes et al. (1983) is that the corresponding extensions u 1 and u 2 are nonnegative in the unit ball of n+ 1 dimensions....
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