The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

By Xavier Ros-Oton, J. Serra. 2012

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Abstract We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω , for some s ∈ ( 0 , 1 ) and g ∈ L ∞ ( Ω ) , then u is C s ( R n ) and u / δ s | Ω is C α up to the boundary ∂Ω for some α ∈ ( 0 , 1 ) , where δ ( x ) = dist ( x , ∂ Ω ) . For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Holder estimates for u and u / δ s . Namely, the C β norms of u and u / δ s in the sets { x ∈ Ω : δ ( x ) ⩾ ρ } are controlled by C ρ s − β and C ρ α − β , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19] , [20] ).

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Citations in the corpus

Semilinear fractional elliptic equations involving measures (2013, background): “1 in [30], for θ ∈ [0,min{α, 1 − α, γ}), there exists C > 0 such that for any r > 0, we have ‖u‖C2α+θ(Ωr) ≤ Cr −α−θ 9 ha l-0 08 20 40 1, v er si on 2 14 M ay 2 01 3”
Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains (2014, methodology): “5 We now would like to pass to the relative error w(t, x) = v(t,x) S(x) − 1, but it turns out to be impossible to write a clean equation due to the nonlocality of L, without passing through the extension technique by Caffarelli-Silvestre [15] in the elliptic case, see also [14, 16, 38] ; as for the ”
The Dirichlet problem for nonlocal operators (2013, background): “Regularity up to the boundary is studied carefully in [22] for the case of the fractional Laplacian.”
Boundary regularity for fully nonlinear integro-differential equations (2014, methodology): “Thus, by standard interior regularity (see for example [42]), we have that [v]C0,1([h,2h]) ≤ C|h|.”
A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, background): “, u ∈ C(R), but u is more regular than f on Ω) [44].”
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity (2013, methodology): “Modifying some arguments in [36] (see also [33]) and applying proposition 2.”
Fractional elliptic equations, Caccioppoli estimates and regularity (2014, background): “Thus in general u is globally in C but not in any C for α > s, see also [22].”
Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces (2016, background): “These two notions of solutions are of course very close, as it is discussed in [RS14] and [SV14], but, since we have a datum f in L2, the weak formulation (1.”
On critical systems involving fractional Laplacian (2017, background): “The regularity theory [7,12] and Proposition 2.”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “The boundary regularity theory for fractional Laplacian, developed in [31,32,37,38], ensures that, for a solution u to (1.”