Nonlocal Harnack inequalities

By Agnese Di Castro, Tuomo Kuusi, Giampiero Palatucci. 2014

289 citations recorded by the source corpus.

Abstract We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c0ced8bb21f065a387d7613d3a515591fd6bd97a

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “In [13], the results of [16, 17] are generalized to functions belonging to a fractional analogue of the so-called De Giorgi classes (we refer to [19, Chapter 7] for the classical definition).”
Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications (2016, unknown): “(1−|x|γ˜) Z ∞ 1 |x| σN−1K(σ)dσ. Using the deﬁnition of γ1, see Lemma 3.1, we can prove that h(x) 6 λfor all x∈ B1(0). Since Lp,su 0 and u>0 in Ω, then using the nonlocal weak Harnack inequality in [11], we get the existence of ε>0 such that u> εin B¯1(0). Therefore we obtain that (3.39) ”
Linear theory for a mixed operator with Neumann conditions (2020, unknown): “We combine for this proof some classical and nonlocal techniques, see e.g. [34, 35,43,45,46,53,54].”
Bifurcation and multiplicity results for critical fractional p‐Laplacian problems (2014, methodology): “Regularity of solutions was obtained in Di Castro, Kuusi and Palatucci [14], [15], Iannizzotto, Mosconi and Squassina [25], Kuusi, Mingione and Sire [27] and Lindgren [28].”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, methodology): “This is one of the novelty with respect to the analog of the definition of supersolutions in the local case, say when s = 1, and it is necessary since here one has to use in a precise way the definition in (10) to deal with the long-range interactions; see Remark 8 below, and also, the regularity es”
Nonlocal Diffusion and Applications (2015, methodology): “We refer to [7, 86, 64] and to the references therein for a detailed introduction to the fractional Harnack Inequality, and to [38] for general estimates of this type.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, methodology): “[41] G. Palatucci, A. Pisante, and Y. Sire, Subcritical approximation of a Yamabe-type nonlocal equation: a gamma- convergence approach, Ann.”
Hölder estimates for viscosity solutions of equations of fractional p-Laplace type (2014, background): “Comments on the equation Let us very briefly point out the difference between the class of equations considered in [10] and [9], and the class of equations considered here when p and s are constant (see also [33] for a similar discussion).”
On the fractional p-Laplacian equations with weight and general datum (2016, methodology): “Z Bj∩{v<lj} dx |x|2β . THE FRACTIONAL P-LAPLACIAN EQUATIONS 35 Thus Z Bj Z Bj w p j|ψj(x) −ψj(y)| dν≤ Cr−ps|Bj∩ {v<lj}|dµ. Now, estimating the term sup{x∈supp ρ j} R IRN\Bρ j dy |x−y|N+ps as in [13] we obtain that (lj− lj+1) p |B j+1 ∩ {v<j+1}|dµ |Bj+1|dµ p p∗ s ≤ C(N,s) r(N−ps−2β) r−ps|B ”